P. 384 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

rfovcnvfvd 38301*

Value of the converse of the operator, ( A O B ), which maps between relations and functions for relations between base sets, A and B, evaluated at function G. (Contributed by RP, 27-Apr-2021.)

|- O = ( a e. _V , b e. _V |-> ( r e. ~P ( a X. b ) |-> ( x e. a |-> { y e. b | x r y } ) ) )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- F = ( A O B )   &    |- ( ph -> G e. ( ~P B ^m A ) )   =>    |- ( ph -> ( `' F ` G ) = { <. x , y >. | ( x e. A /\ y e. ( G ` x ) ) } )

Theorem

fsovd 38302*

Value of the operator, ( A O B ), which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, A and B. (Contributed by RP, 25-Apr-2021.)

|- O = ( a e. _V , b e. _V |-> ( f e. ( ~P b ^m a ) |-> ( y e. b |-> { x e. a | y e. ( f ` x ) } ) ) )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   =>    |- ( ph -> ( A O B ) = ( f e. ( ~P B ^m A ) |-> ( y e. B |-> { x e. A | y e. ( f ` x ) } ) ) )

Theorem

fsovrfovd 38303*

The operator which gives a 1-to-1 a mapping to a subset and a reverse mapping from elements can be composed from the operator which gives a 1-to-1 mapping between relations and functions to subsets and the converse operator. (Contributed by RP, 15-May-2021.)

|- O = ( a e. _V , b e. _V |-> ( f e. ( ~P b ^m a ) |-> ( y e. b |-> { x e. a | y e. ( f ` x ) } ) ) )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- R = ( a e. _V , b e. _V |-> ( r e. ~P ( a X. b ) |-> ( u e. a |-> { v e. b | u r v } ) ) )   &    |- C = ( a e. _V , b e. _V |-> ( s e. ~P ( a X. b ) |-> `' s ) )   =>    |- ( ph -> ( A O B ) = ( ( B R A ) o. ( ( A C B ) o. `' ( A R B ) ) ) )

Theorem

fsovfvd 38304*

Value of the operator, ( A O B ), which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, A and B, when applied to function F. (Contributed by RP, 25-Apr-2021.)

|- O = ( a e. _V , b e. _V |-> ( f e. ( ~P b ^m a ) |-> ( y e. b |-> { x e. a | y e. ( f ` x ) } ) ) )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- G = ( A O B )   &    |- ( ph -> F e. ( ~P B ^m A ) )   =>    |- ( ph -> ( G ` F ) = ( y e. B |-> { x e. A | y e. ( F ` x ) } ) )

Theorem

fsovfvfvd 38305*

Value of the operator, ( A O B ), which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, A and B, when applied to function F and element Y. (Contributed by RP, 25-Apr-2021.)

|- O = ( a e. _V , b e. _V |-> ( f e. ( ~P b ^m a ) |-> ( y e. b |-> { x e. a | y e. ( f ` x ) } ) ) )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- G = ( A O B )   &    |- ( ph -> F e. ( ~P B ^m A ) )   &    |- H = ( G ` F )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( H ` Y ) = { x e. A | Y e. ( F ` x ) } )

Theorem

fsovfd 38306*

The operator, ( A O B ), which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, A and B, gives a function between two sets of functions. (Contributed by RP, 27-Apr-2021.)

|- O = ( a e. _V , b e. _V |-> ( f e. ( ~P b ^m a ) |-> ( y e. b |-> { x e. a | y e. ( f ` x ) } ) ) )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- G = ( A O B )   =>    |- ( ph -> G : ( ~P B ^m A ) --> ( ~P A ^m B ) )

Theorem

fsovcnvlem 38307*

The O operator, which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, gives a family of functions that include their own inverse. (Contributed by RP, 27-Apr-2021.)

|- O = ( a e. _V , b e. _V |-> ( f e. ( ~P b ^m a ) |-> ( y e. b |-> { x e. a | y e. ( f ` x ) } ) ) )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- G = ( A O B )   &    |- H = ( B O A )   =>    |- ( ph -> ( H o. G ) = ( _I |` ( ~P B ^m A ) ) )

Theorem

fsovcnvd 38308*

The value of the converse ( A O B ) is ( B O A ), where O is the operator which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, gives a family of functions that include their own inverse. (Contributed by RP, 27-Apr-2021.)

|- O = ( a e. _V , b e. _V |-> ( f e. ( ~P b ^m a ) |-> ( y e. b |-> { x e. a | y e. ( f ` x ) } ) ) )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- G = ( A O B )   &    |- H = ( B O A )   =>    |- ( ph -> `' G = H )

Theorem

fsovcnvfvd 38309*

The value of the converse of ( A O B ), where O is the operator which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, evaluated at function F. (Contributed by RP, 27-Apr-2021.)

|- O = ( a e. _V , b e. _V |-> ( f e. ( ~P b ^m a ) |-> ( y e. b |-> { x e. a | y e. ( f ` x ) } ) ) )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- G = ( A O B )   &    |- ( ph -> F e. ( ~P A ^m B ) )   =>    |- ( ph -> ( `' G ` F ) = ( y e. A |-> { x e. B | y e. ( F ` x ) } ) )

Theorem

fsovf1od 38310*

The value of ( A O B ) is a bijection, where O is the operator which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets. (Contributed by RP, 27-Apr-2021.)

|- O = ( a e. _V , b e. _V |-> ( f e. ( ~P b ^m a ) |-> ( y e. b |-> { x e. a | y e. ( f ` x ) } ) ) )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- G = ( A O B )   =>    |- ( ph -> G : ( ~P B ^m A ) -1-1-onto-> ( ~P A ^m B ) )

Theorem

dssmapfvd 38311*

Value of the duality operator for self-mappings of subsets of a base set, B. (Contributed by RP, 19-Apr-2021.)

|- O = ( b e. _V |-> ( f e. ( ~P b ^m ~P b ) |-> ( s e. ~P b |-> ( b \ ( f ` ( b \ s ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> B e. V )   =>    |- ( ph -> D = ( f e. ( ~P B ^m ~P B ) |-> ( s e. ~P B |-> ( B \ ( f ` ( B \ s ) ) ) ) ) )

Theorem

dssmapfv2d 38312*

Value of the duality operator for self-mappings of subsets of a base set, B when applied to function F. (Contributed by RP, 19-Apr-2021.)

|- O = ( b e. _V |-> ( f e. ( ~P b ^m ~P b ) |-> ( s e. ~P b |-> ( b \ ( f ` ( b \ s ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> B e. V )   &    |- ( ph -> F e. ( ~P B ^m ~P B ) )   &    |- G = ( D ` F )   =>    |- ( ph -> G = ( s e. ~P B |-> ( B \ ( F ` ( B \ s ) ) ) ) )

Theorem

dssmapfv3d 38313*

Value of the duality operator for self-mappings of subsets of a base set, B when applied to function F and subset S. (Contributed by RP, 19-Apr-2021.)

|- O = ( b e. _V |-> ( f e. ( ~P b ^m ~P b ) |-> ( s e. ~P b |-> ( b \ ( f ` ( b \ s ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> B e. V )   &    |- ( ph -> F e. ( ~P B ^m ~P B ) )   &    |- G = ( D ` F )   &    |- ( ph -> S e. ~P B )   &    |- T = ( G ` S )   =>    |- ( ph -> T = ( B \ ( F ` ( B \ S ) ) ) )

Theorem

dssmapnvod 38314*

For any base set B the duality operator for self-mappings of subsets of that base set is its own inverse, an involution. (Contributed by RP, 20-Apr-2021.)

|- O = ( b e. _V |-> ( f e. ( ~P b ^m ~P b ) |-> ( s e. ~P b |-> ( b \ ( f ` ( b \ s ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> B e. V )   =>    |- ( ph -> `' D = D )

Theorem

dssmapf1od 38315*

For any base set B the duality operator for self-mappings of subsets of that base set is one-to-one and onto. (Contributed by RP, 21-Apr-2021.)

|- O = ( b e. _V |-> ( f e. ( ~P b ^m ~P b ) |-> ( s e. ~P b |-> ( b \ ( f ` ( b \ s ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> B e. V )   =>    |- ( ph -> D : ( ~P B ^m ~P B ) -1-1-onto-> ( ~P B ^m ~P B ) )

Theorem

dssmap2d 38316*

For any base set B the duality operator for self-mappings of subsets of that base set when composed with itself is the restricted identity operator. (Contributed by RP, 21-Apr-2021.)

|- O = ( b e. _V |-> ( f e. ( ~P b ^m ~P b ) |-> ( s e. ~P b |-> ( b \ ( f ` ( b \ s ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> B e. V )   =>    |- ( ph -> ( D o. D ) = ( _I |` ( ~P B ^m ~P B ) ) )

20.27.4.2  Generic Pseudoclosure Spaces, Pseudointeror Spaces, and Pseudoneighborhoods

For any base set, B, an arbitrary mapping of subsets to subsets can be called a pseudoclosure (pseudointerior) function, K, with its dual of a pseudointerior (pseudoclosure), I, related by the involution in dssmapfvd 38311. As K gains properties of the closure (interior) function of a topology on B, so does its dual gain corresponding properties of the interior (closure) function of that topology.

As ( ~P B ^m ~P B ) ~~ ( ~P ~P B ^m B ) there is also a natural isomorphism which maps from I to N (and likewise for K and M, introduced below) which identically gains the properties of the neighborhood function of a topology (modified and restricted to operate on single points). A function dual to N, which Stadler and Stadler refer to as a convergent function, is represented by M in this section.

Based on this and the early treatment of topology in Seifert and Threlfall, it seems reasonable to define a pseudotopology as defined in terms of its base set and one of these functions with theorems treating the equivalence of the other definitions and adding topological structure if enough properties hold true.

	Neighborhoods		Interior		Closure		Convergents	Theorems
Functions	N e. ( ~P ~P B ^m B )		I e. ( ~P B ^m ~P B )		K e. ( ~P B ^m ~P B )		M e. ( ~P ~P B ^m B )
Correspondences (assuming ( X e. B /\ S e. ~P B ))	S e. ( N ` X )	<->	X e. ( I ` S )	<->	-. X e. ( K ` ( B \ S ) )	<->	-. ( B \ S ) e. ( M ` X )	ntrclselnel1 38355, ntrneiel 38379, neicvgel1 38417
	-. ( B \ S ) e. ( N ` X )	<->	-. X e. ( I ` ( B \ S ) )	<->	X e. ( K ` S )	<->	S e. ( M ` X )
Neighborhoods	( N ` X )	=	{ s e. ~P B | X e. ( I ` s ) }	=	{ s e. ~P B | -. X e. ( K ` ( B \ s ) ) }	=	{ s e. ~P B | -. ( B \ s ) e. ( M ` X ) }	ntrneifv3 38380, clsneifv3 38408, neicvgfv 38419
Interior	{ x e. B | S e. ( N ` x ) }	=	( I ` S )	=	( B \ ( K ` ( B \ S ) ) )	=	{ x e. B | -. ( B \ S ) e. ( M ` x ) }	ntrneifv4 38383, ntrclsfv 38357, clsneifv4 38409
Closure	{ x e. B | -. ( B \ S ) e. ( N ` x ) }	=	( B \ ( I ` ( B \ S ) ) )	=	( K ` S )	=	{ x e. B | S e. ( M ` x ) }	clsneifv4 38409, ntrclsfv 38357, ntrneifv4 38383
Convergents	{ s e. ~P B | -. ( B \ s ) e. ( N ` X ) }	=	{ s e. ~P B | -. X e. ( I ` ( B \ s ) ) }	=	{ s e. ~P B | X e. ( K ` s ) }	=	( M ` X )	neicvgfv 38419, clsneifv3 38408, ntrneifv3 38380

We have the following table of equivalences to axioms largely established by Kuratowski. In the formulas in this table, to reduce the width of the columns, if any of the variables x, s, or t are used, then they are implicitly universally quantified and x (respectively s and t) ranges over B (respectively ~P B and ~P B).

Assuming a prefix of: A. x e. B A. s e. ~P B A. t e. ~P B	Neighborhoods	Interior	Closure	Convergents	Equivalence Theorems
K0' Neighborhoods are non-empty.	( N ` x ) =/= (/)	E. u e. ~P B x e. ( I ` u )	E. u e. ~P B -. x e. ( K ` u )	( M ` x ) =/= ~P B	ntrclsneine0 38363, ntrneineine0 38385, ntrneineine1 38386
KA' No neighborhood is equal to the full powerset.	( N ` x ) =/= ~P B	E. u e. ~P B -. x e. ( I ` u )	E. u e. ~P B x e. ( K ` u )	( M ` x ) =/= (/)	ntrclsneine0 38363, ntrneineine0 38385, ntrneineine1 38386
K0 Preservation of the Nullary Union of Closures	B e. ( N ` x )	( I ` B ) = B	( K ` (/) ) = (/)	-. (/) e. ( M ` x )	ntrclscls00 38364, ntrneicls00 38387, ntrneicls11 38388
KA Preservation of the Nullary Union of Interiors	-. (/) e. ( N ` x )	( I ` (/) ) = (/)	( K ` B ) = B	B e. ( M ` x )	ntrclscls00 38364, ntrneicls00 38387, ntrneicls11 38388
K1 Isotonic Montonic	( ( s e. ( N ` x ) /\ s C_ t ) -> t e. ( N ` x ) )	( s C_ t -> ( I ` s ) C_ ( I ` t ) ) — or — ( ( I ` s ) u. ( I ` t ) ) C_ ( I ` ( s u. t ) ) — or — ( I ` ( s i^i t ) ) C_ ( ( I ` s ) i^i ( I ` t ) )	( s C_ t -> ( K ` s ) C_ ( K ` t ) ) — or — ( ( K ` s ) u. ( K ` t ) ) C_ ( K ` ( s u. t ) ) — or — ( K ` ( s i^i t ) ) C_ ( ( K ` s ) i^i ( K ` t ) )	( ( s e. ( M ` x ) /\ s C_ t ) -> t e. ( M ` x ) )	isotone1 38346, isotone2 38347, ntrclsiso 38365, ntrneiiso 38389
K2 Closure is Expansive	( s e. ( N ` x ) -> x e. s )	( I ` s ) C_ s	s C_ ( K ` s )	( x e. s -> s e. ( M ` x ) )	ntrclsk2 38366, ntrneik2 38390, ntrneix2 38391
KB Non-disjoint Neighborhoods	( ( s e. ( N ` x ) /\ t e. ( N ` x ) ) -> ( s i^i t ) =/= (/) )	( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) )	( ( s u. t ) = B -> ( ( K ` s ) u. ( K ` t ) ) = B )	( ( s u. t ) = B -> ( s e. ( M ` x ) \/ t e. ( M ` x ) ) )	ntrclskb 38367, ntrneikb 38392, ntrneixb 38393
K3 Closure is Sub-linear	( ( s e. ( N ` x ) /\ t e. ( N ` x ) ) -> ( s i^i t ) e. ( N ` x ) )	( ( I ` s ) i^i ( I ` t ) ) C_ ( I ` ( s i^i t ) )	( K ` ( s u. t ) ) C_ ( ( K ` s ) u. ( K ` t ) )	( ( s u. t ) e. ( M ` x ) -> ( s e. ( M ` x ) \/ t e. ( M ` x ) ) )	ntrclsk3 38368, ntrneik3 38394, ntrneix3 38395
K13 Closure is finitely linear	( ( s i^i t ) e. ( N ` x ) <-> ( s e. ( N ` x ) /\ t e. ( N ` x ) ) )	( I ` ( s i^i t ) ) = ( ( I ` s ) i^i ( I ` t ) )	( K ` ( s u. t ) ) = ( ( K ` s ) u. ( K ` t ) )	( ( s u. t ) e. ( M ` x ) <-> ( s e. ( M ` x ) \/ t e. ( M ` x ) ) )	ntrclsk13 38369, ntrneik13 38396, ntrneix13 38397
K4 Closure is idempotent	( s e. ( N ` x ) <-> E. u e. ( N ` x ) A. y e. B ( y e. u <-> s e. ( N ` y ) ) )	( I ` ( I ` s ) ) = ( I ` s )	( K ` ( K ` s ) ) = ( K ` s )	( s e. ( M ` x ) <-> E. u e. ( M ` x ) A. y e. B ( y e. u <-> s e. ( M ` y ) ) )	ntrclsk4 38370, ntrneik4 38399

Using these properties as axiomic constraints on the functions, certain collections of them give rise to named spaces.

Space	Foundational Axioms	Derived Axioms	Theorems
Csázár Generalized Neighborhood Space	K2	KA', KA, KB	ntrk2imkb 38335, ntrkbimka 38336, neik0imk0p 38334
Min Strong Generalized Neighborhood Space	K2, K3	KA', KA, KB	ntrk2imkb 38335, ntrkbimka 38336, neik0imk0p 38334
Gniłka Extended Topology	K0', K1	K0	neik0pk1imk0 38345
Brissaud Space	K0, K2	K0', KA', KA, KB	neik0imk0p 38334, ntrk2imkb 38335, ntrkbimka 38336
Neighborhood Space	K0', K1, K2	K0, KA', KA, KB	neik0pk1imk0 38345, ntrk2imkb 38335, ntrkbimka 38336, neik0imk0p 38334
Davey and Priestley Intersection Structure	K1, K4
Moore Closure Space	K1, K2, K4	KA', KA, KB	ntrk2imkb 38335, ntrkbimka 38336, neik0imk0p 38334
Convex Closure Space	K0', K1, K2, K4	K0, KA', KA, KB	neik0pk1imk0 38345, ntrk2imkb 38335, ntrkbimka 38336, neik0imk0p 38334
Smyth Neighborhood Space	K0', K13	K0, K1, K3	neik0pk1imk0 38345, ntrk1k3eqk13 38348
Čech Closure Space Pretopological Space	K0', K2, K13	K0, K1, KA', KA, KB, K3	neik0pk1imk0 38345, ntrk2imkb 38335, ntrkbimka 38336, neik0imk0p 38334, ntrk1k3eqk13 38348
Topological Space	K0', K2, K13, K4	K0, K1, KA', KA, KB, K3	neik0pk1imk0 38345, ntrk2imkb 38335, ntrkbimka 38336, neik0imk0p 38334, ntrk1k3eqk13 38348
Alexandroff Space	K0', K2, K5	K0, K1, KA', KA, KB, K3, K13	neik0pk1imk0 38345, ntrk2imkb 38335, ntrkbimka 38336, neik0imk0p 38334, ntrk1k3eqk13 38348, TBD
Alexandroff Topological Space	K0', K2, K4, K5	K0, K1, KA', KA, KB, K3, K13	neik0pk1imk0 38345, ntrk2imkb 38335, ntrkbimka 38336, neik0imk0p 38334, ntrk1k3eqk13 38348, TBD

Theorem

sscon34b 38317

Relative complementation reverses inclusion of subclasses. Relativized version of complss 3751. (Contributed by RP, 3-Jun-2021.)

|- ( ( A C_ C /\ B C_ C ) -> ( A C_ B <-> ( C \ B ) C_ ( C \ A ) ) )

Theorem

rcompleq 38318

Two subclasses are equal if and only if their relative complements are equal. Relativized version of compleq 3752. (Contributed by RP, 10-Jun-2021.)

|- ( ( A C_ C /\ B C_ C ) -> ( A = B <-> ( C \ A ) = ( C \ B ) ) )

Theorem

or3or 38319

Decompose disjunction into three cases. (Contributed by RP, 5-Jul-2021.)

|- ( ( ph \/ ps ) <-> ( ( ph /\ ps ) \/ ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) )

Theorem

andi3or 38320

Distribute over triple disjunction. (Contributed by RP, 5-Jul-2021.)

|- ( ( ph /\ ( ps \/ ch \/ th ) ) <-> ( ( ph /\ ps ) \/ ( ph /\ ch ) \/ ( ph /\ th ) ) )

Theorem

uneqsn 38321

If a union of classes is equal to a singleton then at least one class is equal to the singleton while the other may be equal to the empty set. (Contributed by RP, 5-Jul-2021.)

|- ( ( A u. B ) = { C } <-> ( ( A = { C } /\ B = { C } ) \/ ( A = { C } /\ B = (/) ) \/ ( A = (/) /\ B = { C } ) ) )

Theorem

df3o2 38322

Ordinal 3 is the triplet containing ordinals 0, 1 and 2. (Contributed by RP, 8-Jul-2021.)

|- 3o = { (/) , 1o , 2o }

Theorem

df3o3 38323

Ordinal 3 , fully expanded. (Contributed by RP, 8-Jul-2021.)

|- 3o = { (/) , { (/) } , { (/) , { (/) } } }

Theorem

brfvimex 38324

If a binary relation holds and the relation is the value of a function, then the argument to that function is a set. (Contributed by RP, 22-May-2021.)

|- ( ph -> A R B )   &    |- ( ph -> R = ( F ` C ) )   =>    |- ( ph -> C e. _V )

Theorem

brovmptimex 38325*

If a binary relation holds and the relation is the value of a binary operation built with maps-to, then the arguments to that operation are sets. (Contributed by RP, 22-May-2021.)

|- F = ( x e. E , y e. G |-> H )   &    |- ( ph -> A R B )   &    |- ( ph -> R = ( C F D ) )   =>    |- ( ph -> ( C e. _V /\ D e. _V ) )

Theorem

brovmptimex1 38326*

If a binary relation holds and the relation is the value of a binary operation built with maps-to, then the arguments to that operation are sets. (Contributed by RP, 22-May-2021.)

|- F = ( x e. E , y e. G |-> H )   &    |- ( ph -> A R B )   &    |- ( ph -> R = ( C F D ) )   =>    |- ( ph -> C e. _V )

Theorem

brovmptimex2 38327*

If a binary relation holds and the relation is the value of a binary operation built with maps-to, then the arguments to that operation are sets. (Contributed by RP, 22-May-2021.)

|- F = ( x e. E , y e. G |-> H )   &    |- ( ph -> A R B )   &    |- ( ph -> R = ( C F D ) )   =>    |- ( ph -> D e. _V )

Theorem

brcoffn 38328

Conditions allowing the decomposition of a binary relation. (Contributed by RP, 7-Jun-2021.)

|- ( ph -> C Fn Y )   &    |- ( ph -> D : X --> Y )   &    |- ( ph -> A ( C o. D ) B )   =>    |- ( ph -> ( A D ( D ` A ) /\ ( D ` A ) C B ) )

Theorem

brcofffn 38329

Conditions allowing the decomposition of a binary relation. (Contributed by RP, 8-Jun-2021.)

|- ( ph -> C Fn Z )   &    |- ( ph -> D : Y --> Z )   &    |- ( ph -> E : X --> Y )   &    |- ( ph -> A ( C o. ( D o. E ) ) B )   =>    |- ( ph -> ( A E ( E ` A ) /\ ( E ` A ) D ( D ` ( E ` A ) ) /\ ( D ` ( E ` A ) ) C B ) )

Theorem

brco2f1o 38330

Conditions allowing the decomposition of a binary relation. (Contributed by RP, 8-Jun-2021.)

|- ( ph -> C : Y -1-1-onto-> Z )   &    |- ( ph -> D : X -1-1-onto-> Y )   &    |- ( ph -> A ( C o. D ) B )   =>    |- ( ph -> ( ( `' C ` B ) C B /\ A D ( `' C ` B ) ) )

Theorem

brco3f1o 38331

Conditions allowing the decomposition of a binary relation. (Contributed by RP, 8-Jun-2021.)

|- ( ph -> C : Y -1-1-onto-> Z )   &    |- ( ph -> D : X -1-1-onto-> Y )   &    |- ( ph -> E : W -1-1-onto-> X )   &    |- ( ph -> A ( C o. ( D o. E ) ) B )   =>    |- ( ph -> ( ( `' C ` B ) C B /\ ( `' D ` ( `' C ` B ) ) D ( `' C ` B ) /\ A E ( `' D ` ( `' C ` B ) ) ) )

Theorem

ntrclsbex 38332

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then the base set exists. (Contributed by RP, 21-May-2021.)

|- D = ( O ` B )   &    |- ( ph -> I D K )   =>    |- ( ph -> B e. _V )

Theorem

ntrclsrcomplex 38333

The relative complement of the class S exists as a subset of the base set. (Contributed by RP, 25-Jun-2021.)

|- D = ( O ` B )   &    |- ( ph -> I D K )   =>    |- ( ph -> ( B \ S ) e. ~P B )

Theorem

neik0imk0p 38334

Kuratowski's K0 axiom implies K0'. Neighborhood version. Also a proof the dual KA axiom imples KA' when considering the convergents. (Contributed by RP, 28-Jun-2021.)

|- ( A. x e. B B e. ( N ` x ) -> A. x e. B ( N ` x ) =/= (/) )

Theorem

ntrk2imkb 38335*

If an interior function is contracting, the interiors of disjoint sets are disjoint. Kuratowski's K2 axiom implies KB. Interior version. (Contributed by RP, 9-Jun-2021.)

|- ( A. s e. ~P B ( I ` s ) C_ s -> A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) )

Theorem

ntrkbimka 38336*

If the interiors of disjoint sets are disjoint, then the interior of the empty set is the empty set. (Contributed by RP, 14-Jun-2021.)

|- ( A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) -> ( I ` (/) ) = (/) )

Theorem

ntrk0kbimka 38337*

If the interiors of disjoint sets are disjoint and the interior of the base set is the base set, then the interior of the empty set is the empty set. Obsolete version of ntrkbimka 38336. (Contributed by RP, 12-Jun-2021.)

|- ( ( B e. V /\ I e. ( ~P B ^m ~P B ) ) -> ( ( ( I ` B ) = B /\ A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) ) -> ( I ` (/) ) = (/) ) )

Theorem

clsk3nimkb 38338*

If the base set is not empty, axiom K3 does not imply KB. An concrete example with a pseudo-closure function of k = ( x e. ~P b |-> ( b \ x ) ) is given. (Contributed by RP, 16-Jun-2021.)

|- -. A. b A. k e. ( ~P b ^m ~P b ) ( A. s e. ~P b A. t e. ~P b ( k ` ( s u. t ) ) C_ ( ( k ` s ) u. ( k ` t ) ) -> A. s e. ~P b A. t e. ~P b ( ( s u. t ) = b -> ( ( k ` s ) u. ( k ` t ) ) = b ) )

Theorem

clsk1indlem0 38339

The ansatz closure function ( r e. ~P 3o |-> if ( r = { (/) } , { (/) , 1o } , r ) ) has the K0 property of preserving the nullary union. (Contributed by RP, 6-Jul-2021.)

|- K = ( r e. ~P 3o |-> if ( r = { (/) } , { (/) , 1o } , r ) )   =>    |- ( K ` (/) ) = (/)

Theorem

clsk1indlem2 38340*

The ansatz closure function ( r e. ~P 3o |-> if ( r = { (/) } , { (/) , 1o } , r ) ) has the K2 property of expanding. (Contributed by RP, 6-Jul-2021.)

|- K = ( r e. ~P 3o |-> if ( r = { (/) } , { (/) , 1o } , r ) )   =>    |- A. s e. ~P 3o s C_ ( K ` s )

Theorem

clsk1indlem3 38341*

The ansatz closure function ( r e. ~P 3o |-> if ( r = { (/) } , { (/) , 1o } , r ) ) has the K3 property of being sub-linear. (Contributed by RP, 6-Jul-2021.)

|- K = ( r e. ~P 3o |-> if ( r = { (/) } , { (/) , 1o } , r ) )   =>    |- A. s e. ~P 3o A. t e. ~P 3o ( K ` ( s u. t ) ) C_ ( ( K ` s ) u. ( K ` t ) )

Theorem

clsk1indlem4 38342*

The ansatz closure function ( r e. ~P 3o |-> if ( r = { (/) } , { (/) , 1o } , r ) ) has the K4 property of idempotence. (Contributed by RP, 6-Jul-2021.)

|- K = ( r e. ~P 3o |-> if ( r = { (/) } , { (/) , 1o } , r ) )   =>    |- A. s e. ~P 3o ( K ` ( K ` s ) ) = ( K ` s )

Theorem

clsk1indlem1 38343*

The ansatz closure function ( r e. ~P 3o |-> if ( r = { (/) } , { (/) , 1o } , r ) ) does not have the K1 property of isotony. (Contributed by RP, 6-Jul-2021.)

|- K = ( r e. ~P 3o |-> if ( r = { (/) } , { (/) , 1o } , r ) )   =>    |- E. s e. ~P 3o E. t e. ~P 3o ( s C_ t /\ -. ( K ` s ) C_ ( K ` t ) )

Theorem

clsk1independent 38344*

For generalized closure functions, property K1 (isotony) is independent of the properties K0, K2, K3, K4. This contradicts a claim which appears in preprints of Table 2 in Bärbel M. R. Stadler and Peter F. Stadler. "Generalized Topological Spaces in Evolutionary Theory and Combinatorial Chemistry." J. Chem. Inf. Comput. Sci., 42:577-585, 2002. Proceedings MCC 2001, Dubrovnik. The same table row implying K1 follows from the other four appears in the supplemental materials Bärbel M. R. Stadler and Peter F. Stadler. "Basic Properties of Closure Spaces" 2001 on page 12. (Contributed by RP, 5-Jul-2021.)

|- ( ph <-> ( k ` (/) ) = (/) )   &    |- ( ps <-> A. s e. ~P b A. t e. ~P b ( s C_ t -> ( k ` s ) C_ ( k ` t ) ) )   &    |- ( ch <-> A. s e. ~P b s C_ ( k ` s ) )   &    |- ( th <-> A. s e. ~P b A. t e. ~P b ( k ` ( s u. t ) ) C_ ( ( k ` s ) u. ( k ` t ) ) )   &    |- ( ta <-> A. s e. ~P b ( k ` ( k ` s ) ) = ( k ` s ) )   =>    |- -. A. b A. k e. ( ~P b ^m ~P b ) ( ( ( ph /\ ch ) /\ ( th /\ ta ) ) -> ps )

Theorem

neik0pk1imk0 38345*

Kuratowski's K0' and K1 axioms imply K0. Neighborhood version. (Contributed by RP, 3-Jun-2021.)

|- ( ph -> B e. V )   &    |- ( ph -> N e. ( ~P ~P B ^m B ) )   &    |- ( ph -> A. x e. B ( N ` x ) =/= (/) )   &    |- ( ph -> A. x e. B A. s e. ~P B A. t e. ~P B ( ( s e. ( N ` x ) /\ s C_ t ) -> t e. ( N ` x ) ) )   =>    |- ( ph -> A. x e. B B e. ( N ` x ) )

Theorem

isotone1 38346*

Two different ways to say subset relation persists across applications of a function. (Contributed by RP, 31-May-2021.)

|- ( A. a e. ~P A A. b e. ~P A ( a C_ b -> ( F ` a ) C_ ( F ` b ) ) <-> A. a e. ~P A A. b e. ~P A ( ( F ` a ) u. ( F ` b ) ) C_ ( F ` ( a u. b ) ) )

Theorem

isotone2 38347*

Two different ways to say subset relation persists across applications of a function. (Contributed by RP, 31-May-2021.)

|- ( A. a e. ~P A A. b e. ~P A ( a C_ b -> ( F ` a ) C_ ( F ` b ) ) <-> A. a e. ~P A A. b e. ~P A ( F ` ( a i^i b ) ) C_ ( ( F ` a ) i^i ( F ` b ) ) )

Theorem

ntrk1k3eqk13 38348*

An interior function is both monotone and sub-linear if and only if it is finitely linear. (Contributed by RP, 18-Jun-2021.)

|- ( ( A. s e. ~P B A. t e. ~P B ( s C_ t -> ( I ` s ) C_ ( I ` t ) ) /\ A. s e. ~P B A. t e. ~P B ( ( I ` s ) i^i ( I ` t ) ) C_ ( I ` ( s i^i t ) ) ) <-> A. s e. ~P B A. t e. ~P B ( I ` ( s i^i t ) ) = ( ( I ` s ) i^i ( I ` t ) ) )

Theorem

ntrclsf1o 38349*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator we may characterize the relation as part of a 1-to-1 onto function. (Contributed by RP, 29-May-2021.)

|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> I D K )   =>    |- ( ph -> D : ( ~P B ^m ~P B ) -1-1-onto-> ( ~P B ^m ~P B ) )

Theorem

ntrclsnvobr 38350*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then they are related the opposite way. (Contributed by RP, 21-May-2021.)

|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> I D K )   =>    |- ( ph -> K D I )

Theorem

ntrclsiex 38351*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then those functions are maps of subsets to subsets. (Contributed by RP, 21-May-2021.)

|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> I D K )   =>    |- ( ph -> I e. ( ~P B ^m ~P B ) )

Theorem

ntrclskex 38352*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then those functions are maps of subsets to subsets. (Contributed by RP, 21-May-2021.)

|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> I D K )   =>    |- ( ph -> K e. ( ~P B ^m ~P B ) )

Theorem

ntrclsfv1 38353*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is a functional relation between them (Contributed by RP, 28-May-2021.)

|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> I D K )   =>    |- ( ph -> ( D ` I ) = K )

Theorem

ntrclsfv2 38354*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is a functional relation between them (Contributed by RP, 28-May-2021.)

|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> I D K )   =>    |- ( ph -> ( D ` K ) = I )

Theorem

ntrclselnel1 38355*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is an equivalence between membership in the interior of a set and non-membership in the closure of the complement of the set. (Contributed by RP, 28-May-2021.)

|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> I D K )   &    |- ( ph -> X e. B )   &    |- ( ph -> S e. ~P B )   =>    |- ( ph -> ( X e. ( I ` S ) <-> -. X e. ( K ` ( B \ S ) ) ) )

Theorem

ntrclselnel2 38356*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is an equivalence between membership in interior of the complement of a set and non-membership in the closure of the set. (Contributed by RP, 28-May-2021.)

|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> I D K )   &    |- ( ph -> X e. B )   &    |- ( ph -> S e. ~P B )   =>    |- ( ph -> ( X e. ( I ` ( B \ S ) ) <-> -. X e. ( K ` S ) ) )

Theorem

ntrclsfv 38357*

The value of the interior (closure) expressed in terms of the closure (interior). (Contributed by RP, 25-Jun-2021.)

|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> I D K )   &    |- ( ph -> S e. ~P B )   =>    |- ( ph -> ( I ` S ) = ( B \ ( K ` ( B \ S ) ) ) )

Theorem

ntrclsfveq1 38358*

If interior and closure functions are related then specific function values are complementary. (Contributed by RP, 27-Jun-2021.)

|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> I D K )   &    |- ( ph -> S e. ~P B )   &    |- ( ph -> C e. ~P B )   =>    |- ( ph -> ( ( I ` S ) = C <-> ( K ` ( B \ S ) ) = ( B \ C ) ) )

Theorem

ntrclsfveq2 38359*

If interior and closure functions are related then specific function values are complementary. (Contributed by RP, 27-Jun-2021.)

|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> I D K )   &    |- ( ph -> S e. ~P B )   &    |- ( ph -> C e. ~P B )   =>    |- ( ph -> ( ( I ` ( B \ S ) ) = C <-> ( K ` S ) = ( B \ C ) ) )

Theorem

ntrclsfveq 38360*

If interior and closure functions are related then equality of a pair of function values is equivalent to equality of a pair of the other function's values. (Contributed by RP, 27-Jun-2021.)

|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> I D K )   &    |- ( ph -> S e. ~P B )   &    |- ( ph -> T e. ~P B )   =>    |- ( ph -> ( ( I ` S ) = ( I ` T ) <-> ( K ` ( B \ S ) ) = ( K ` ( B \ T ) ) ) )

Theorem

ntrclsss 38361*

If interior and closure functions are related then a subset relation of a pair of function values is equivalent to subset relation of a pair of the other function's values. (Contributed by RP, 27-Jun-2021.)

|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> I D K )   &    |- ( ph -> S e. ~P B )   &    |- ( ph -> T e. ~P B )   =>    |- ( ph -> ( ( I ` S ) C_ ( I ` T ) <-> ( K ` ( B \ T ) ) C_ ( K ` ( B \ S ) ) ) )

Theorem

ntrclsneine0lem 38362*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that at least one (pseudo-)neighborbood of a particular point exists hold equally. (Contributed by RP, 21-May-2021.)

|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> I D K )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( E. s e. ~P B X e. ( I ` s ) <-> E. s e. ~P B -. X e. ( K ` s ) ) )

Theorem

ntrclsneine0 38363*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that for every point, at least one (pseudo-)neighborbood exists hold equally. (Contributed by RP, 21-May-2021.)

|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> I D K )   =>    |- ( ph -> ( A. x e. B E. s e. ~P B x e. ( I ` s ) <-> A. x e. B E. s e. ~P B -. x e. ( K ` s ) ) )

Theorem

ntrclscls00 38364*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 1-Jun-2021.)

|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> I D K )   =>    |- ( ph -> ( ( I ` B ) = B <-> ( K ` (/) ) = (/) ) )

Theorem

ntrclsiso 38365*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that either is isotonic hold equally. (Contributed by RP, 3-Jun-2021.)

|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> I D K )   =>    |- ( ph -> ( A. s e. ~P B A. t e. ~P B ( s C_ t -> ( I ` s ) C_ ( I ` t ) ) <-> A. s e. ~P B A. t e. ~P B ( s C_ t -> ( K ` s ) C_ ( K ` t ) ) ) )

Theorem

ntrclsk2 38366*

An interior function is contracting if and only if the closure function is expansive. (Contributed by RP, 9-Jun-2021.)

|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> I D K )   =>    |- ( ph -> ( A. s e. ~P B ( I ` s ) C_ s <-> A. s e. ~P B s C_ ( K ` s ) ) )

Theorem

ntrclskb 38367*

The interiors of disjoint sets are disjoint if and only if the closures of sets that span the base set also span the base set. (Contributed by RP, 10-Jun-2021.)

|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> I D K )   =>    |- ( ph -> ( A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) <-> A. s e. ~P B A. t e. ~P B ( ( s u. t ) = B -> ( ( K ` s ) u. ( K ` t ) ) = B ) ) )

Theorem

ntrclsk3 38368*

The intersection of interiors of a every pair is a subset of the interior of the intersection of the pair if an only if the closure of the union of every pair is a subset of the union of closures of the pair. (Contributed by RP, 19-Jun-2021.)

|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> I D K )   =>    |- ( ph -> ( A. s e. ~P B A. t e. ~P B ( ( I ` s ) i^i ( I ` t ) ) C_ ( I ` ( s i^i t ) ) <-> A. s e. ~P B A. t e. ~P B ( K ` ( s u. t ) ) C_ ( ( K ` s ) u. ( K ` t ) ) ) )

Theorem

ntrclsk13 38369*

The interior of the intersection of any pair is equal to the intersection of the interiors if and only if the closure of the unions of any pair is equal to the union of closures. (Contributed by RP, 19-Jun-2021.)

|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> I D K )   =>    |- ( ph -> ( A. s e. ~P B A. t e. ~P B ( I ` ( s i^i t ) ) = ( ( I ` s ) i^i ( I ` t ) ) <-> A. s e. ~P B A. t e. ~P B ( K ` ( s u. t ) ) = ( ( K ` s ) u. ( K ` t ) ) ) )

Theorem

ntrclsk4 38370*

Idempotence of the interior function is equivalent to idempotence of the closure function. (Contributed by RP, 10-Jul-2021.)

|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )   &    |- D = ( O ` B )   &    |- ( ph -> I D K )   =>    |- ( ph -> ( A. s e. ~P B ( I ` ( I ` s ) ) = ( I ` s ) <-> A. s e. ~P B ( K ` ( K ` s ) ) = ( K ` s ) ) )

Theorem

ntrneibex 38371*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F, then the base set exists. (Contributed by RP, 29-May-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> B e. _V )

Theorem

ntrneircomplex 38372*

The relative complement of the class S exists as a subset of the base set. (Contributed by RP, 26-Jun-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> ( B \ S ) e. ~P B )

Theorem

ntrneif1o 38373*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F, we may characterize the relation as part of a 1-to-1 onto function. (Contributed by RP, 29-May-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> F : ( ~P B ^m ~P B ) -1-1-onto-> ( ~P ~P B ^m B ) )

Theorem

ntrneiiex 38374*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F, then the interior function exists. (Contributed by RP, 29-May-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> I e. ( ~P B ^m ~P B ) )

Theorem

ntrneinex 38375*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F, then the neighborhood function exists. (Contributed by RP, 29-May-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> N e. ( ~P ~P B ^m B ) )

Theorem

ntrneicnv 38376*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F, then converse of F is known. (Contributed by RP, 29-May-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> `' F = ( B O ~P B ) )

Theorem

ntrneifv1 38377*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F, then the function value of F is the neighborhood function. (Contributed by RP, 29-May-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> ( F ` I ) = N )

Theorem

ntrneifv2 38378*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F, then the function value of converse of F is the interior function. (Contributed by RP, 29-May-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> ( `' F ` N ) = I )

Theorem

ntrneiel 38379*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F, then there is an equivalence between membership in the interior of a set and non-membership in the closure of the complement of the set. (Contributed by RP, 29-May-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   &    |- ( ph -> X e. B )   &    |- ( ph -> S e. ~P B )   =>    |- ( ph -> ( X e. ( I ` S ) <-> S e. ( N ` X ) ) )

Theorem

ntrneifv3 38380*

The value of the neighbors (convergents) expressed in terms of the interior (closure) function. (Contributed by RP, 26-Jun-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( N ` X ) = { s e. ~P B | X e. ( I ` s ) } )

Theorem

ntrneineine0lem 38381*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F, then conditions equal to claiming that for every point, at least one (pseudo-)neighborbood exists hold equally. (Contributed by RP, 29-May-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( E. s e. ~P B X e. ( I ` s ) <-> ( N ` X ) =/= (/) ) )

Theorem

ntrneineine1lem 38382*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F, then conditions equal to claiming that for every point, at not all subsets are (pseudo-)neighborboods hold equally. (Contributed by RP, 1-Jun-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( E. s e. ~P B -. X e. ( I ` s ) <-> ( N ` X ) =/= ~P B ) )

Theorem

ntrneifv4 38383*

The value of the interior (closure) expressed in terms of the neighbors (convergents) function. (Contributed by RP, 26-Jun-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   &    |- ( ph -> S e. ~P B )   =>    |- ( ph -> ( I ` S ) = { x e. B | S e. ( N ` x ) } )

Theorem

ntrneiel2 38384*

Membership in iterated interior of a set is equivalent to there existing a particular neighborhood of that member such that points are members of that neighborhood if and only if the set is a neighborhood of each of those points. (Contributed by RP, 11-Jul-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   &    |- ( ph -> X e. B )   &    |- ( ph -> S e. ~P B )   =>    |- ( ph -> ( X e. ( I ` ( I ` S ) ) <-> E. u e. ( N ` X ) A. y e. B ( y e. u <-> S e. ( N ` y ) ) ) )

Theorem

ntrneineine0 38385*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F, then conditions equal to claiming that for every point, at least one (pseudo-)neighborbood exists hold equally. (Contributed by RP, 29-May-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> ( A. x e. B E. s e. ~P B x e. ( I ` s ) <-> A. x e. B ( N ` x ) =/= (/) ) )

Theorem

ntrneineine1 38386*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F, then conditions equal to claiming that for every point, at not all subsets are (pseudo-)neighborboods hold equally. (Contributed by RP, 1-Jun-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> ( A. x e. B E. s e. ~P B -. x e. ( I ` s ) <-> A. x e. B ( N ` x ) =/= ~P B ) )

Theorem

ntrneicls00 38387*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F, then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 2-Jun-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> ( ( I ` B ) = B <-> A. x e. B B e. ( N ` x ) ) )

Theorem

ntrneicls11 38388*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F, then conditions equal to claiming that the interior of the empty set is the empty set hold equally. (Contributed by RP, 2-Jun-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> ( ( I ` (/) ) = (/) <-> A. x e. B -. (/) e. ( N ` x ) ) )

Theorem

ntrneiiso 38389*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F, then conditions equal to claiming that the interior function is isotonic hold equally. (Contributed by RP, 3-Jun-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> ( A. s e. ~P B A. t e. ~P B ( s C_ t -> ( I ` s ) C_ ( I ` t ) ) <-> A. x e. B A. s e. ~P B A. t e. ~P B ( ( s e. ( N ` x ) /\ s C_ t ) -> t e. ( N ` x ) ) ) )

Theorem

ntrneik2 38390*

An interior function is contracting if and only if all the neighborhoods of a point contain that point. (Contributed by RP, 11-Jun-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> ( A. s e. ~P B ( I ` s ) C_ s <-> A. x e. B A. s e. ~P B ( s e. ( N ` x ) -> x e. s ) ) )

Theorem

ntrneix2 38391*

An interior (closure) function is expansive if and only if all subsets which contain a point are neighborhoods (convergents) of that point. (Contributed by RP, 11-Jun-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> ( A. s e. ~P B s C_ ( I ` s ) <-> A. x e. B A. s e. ~P B ( x e. s -> s e. ( N ` x ) ) ) )

Theorem

ntrneikb 38392*

The interiors of disjoint sets are disjoint if and only if the neighborhoods of every point contain no disjoint sets. (Contributed by RP, 11-Jun-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> ( A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) <-> A. x e. B A. s e. ~P B A. t e. ~P B ( ( s e. ( N ` x ) /\ t e. ( N ` x ) ) -> ( s i^i t ) =/= (/) ) ) )

Theorem

ntrneixb 38393*

The interiors (closures) of sets that span the base set also span the base set if and only if the neighborhoods (convergents) of every point contain at least one of every pair of sets that span the base set. (Contributed by RP, 11-Jun-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> ( A. s e. ~P B A. t e. ~P B ( ( s u. t ) = B -> ( ( I ` s ) u. ( I ` t ) ) = B ) <-> A. x e. B A. s e. ~P B A. t e. ~P B ( ( s u. t ) = B -> ( s e. ( N ` x ) \/ t e. ( N ` x ) ) ) ) )

Theorem

ntrneik3 38394*

The intersection of interiors of any pair is a subset of the interior of the intersection if and only if the intersection of any two neighborhoods of a point is also a neighborhood. (Contributed by RP, 19-Jun-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> ( A. s e. ~P B A. t e. ~P B ( ( I ` s ) i^i ( I ` t ) ) C_ ( I ` ( s i^i t ) ) <-> A. x e. B A. s e. ~P B A. t e. ~P B ( ( s e. ( N ` x ) /\ t e. ( N ` x ) ) -> ( s i^i t ) e. ( N ` x ) ) ) )

Theorem

ntrneix3 38395*

The closure of the union of any pair is a subset of the union of closures if and only if the union of any pair belonging to the convergents of a point implies at least one of the pair belongs to the the convergents of that point. (Contributed by RP, 19-Jun-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> ( A. s e. ~P B A. t e. ~P B ( I ` ( s u. t ) ) C_ ( ( I ` s ) u. ( I ` t ) ) <-> A. x e. B A. s e. ~P B A. t e. ~P B ( ( s u. t ) e. ( N ` x ) -> ( s e. ( N ` x ) \/ t e. ( N ` x ) ) ) ) )

Theorem

ntrneik13 38396*

The interior of the intersection of any pair equals intersection of interiors if and only if the intersection of any pair belonging to the neighborhood of a point is equivalent to both of the pair belonging to the neighborhood of that point. (Contributed by RP, 19-Jun-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> ( A. s e. ~P B A. t e. ~P B ( I ` ( s i^i t ) ) = ( ( I ` s ) i^i ( I ` t ) ) <-> A. x e. B A. s e. ~P B A. t e. ~P B ( ( s i^i t ) e. ( N ` x ) <-> ( s e. ( N ` x ) /\ t e. ( N ` x ) ) ) ) )

Theorem

ntrneix13 38397*

The closure of the union of any pair is equal to the union of closures if and only if the union of any pair belonging to the convergents of a point if equivalent to at least one of the pain belonging to the convergents of that point. (Contributed by RP, 19-Jun-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> ( A. s e. ~P B A. t e. ~P B ( I ` ( s u. t ) ) = ( ( I ` s ) u. ( I ` t ) ) <-> A. x e. B A. s e. ~P B A. t e. ~P B ( ( s u. t ) e. ( N ` x ) <-> ( s e. ( N ` x ) \/ t e. ( N ` x ) ) ) ) )

Theorem

ntrneik4w 38398*

Idempotence of the interior function is equivalent to saying a set is a neighborhood of a point if and only if the interior of the set is a neighborhood of a point. (Contributed by RP, 11-Jul-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> ( A. s e. ~P B ( I ` ( I ` s ) ) = ( I ` s ) <-> A. x e. B A. s e. ~P B ( s e. ( N ` x ) <-> ( I ` s ) e. ( N ` x ) ) ) )

Theorem

ntrneik4 38399*

Idempotence of the interior function is equivalent to stating a set, s, is a neighborhood of a point, x is equivalent to there existing a special neighborhood, u, of x such that a point is an element of the special neighborhood if and only if s is also a neighborhood of the point. (Contributed by RP, 11-Jul-2021.)

|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )   &    |- F = ( ~P B O B )   &    |- ( ph -> I F N )   =>    |- ( ph -> ( A. s e. ~P B ( I ` ( I ` s ) ) = ( I ` s ) <-> A. x e. B A. s e. ~P B ( s e. ( N ` x ) <-> E. u e. ( N ` x ) A. y e. B ( y e. u <-> s e. ( N ` y ) ) ) ) )

Theorem

clsneibex 38400

If (pseudo-)closure and (pseudo-)neighborhood functions are related by the composite operator, H, then the base set exists. (Contributed by RP, 4-Jun-2021.)

|- D = ( P ` B )   &    |- H = ( F o. D )   &    |- ( ph -> K H N )   =>    |- ( ph -> B e. _V )