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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | distopon 20801 | The discrete topology on a set , with base set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
TopOn | ||
Theorem | sn0topon 20802 | The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
TopOn | ||
Theorem | sn0top 20803 | The singleton of the empty set is a topology. (Contributed by Stefan Allan, 3-Mar-2006.) (Proof shortened by Mario Carneiro, 13-Aug-2015.) |
Theorem | indislem 20804 | A lemma to eliminate some sethood hypotheses when dealing with the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Theorem | indistopon 20805 | The indiscrete topology on a set . Part of Example 2 in [Munkres] p. 77. (Contributed by Mario Carneiro, 13-Aug-2015.) |
TopOn | ||
Theorem | indistop 20806 | The indiscrete topology on a set . Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 16-Jul-2006.) (Revised by Stefan Allan, 6-Nov-2008.) (Revised by Mario Carneiro, 13-Aug-2015.) |
Theorem | indisuni 20807 | The base set of the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Theorem | fctop 20808* | The finite complement topology on a set . Example 3 in [Munkres] p. 77. (Contributed by FL, 15-Aug-2006.) (Revised by Mario Carneiro, 13-Aug-2015.) |
TopOn | ||
Theorem | fctop2 20809* | The finite complement topology on a set . Example 3 in [Munkres] p. 77. (This version of fctop 20808 requires the Axiom of Infinity.) (Contributed by FL, 20-Aug-2006.) |
TopOn | ||
Theorem | cctop 20810* | The countable complement topology on a set . Example 4 in [Munkres] p. 77. (Contributed by FL, 23-Aug-2006.) (Revised by Mario Carneiro, 13-Aug-2015.) |
TopOn | ||
Theorem | ppttop 20811* | The particular point topology. (Contributed by Mario Carneiro, 3-Sep-2015.) |
TopOn | ||
Theorem | pptbas 20812* | The particular point topology is generated by a basis consisting of pairs for each . (Contributed by Mario Carneiro, 3-Sep-2015.) |
Theorem | epttop 20813* | The excluded point topology. (Contributed by Mario Carneiro, 3-Sep-2015.) |
TopOn | ||
Theorem | indistpsx 20814 | The indiscrete topology on a set expressed as a topological space, using explicit structure component references. Compare with indistps 20815 and indistps2 20816. The advantage of this version is that the actual function for the structure is evident, and df-ndx 15860 is not needed, nor any other special definition outside of basic set theory. The disadvantage is that if the indices of the component definitions df-base 15863 and df-tset 15960 are changed in the future, this theorem will also have to be changed. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use indistps 20815 instead. (New usage is discouraged.) (Contributed by FL, 19-Jul-2006.) |
Theorem | indistps 20815 | The indiscrete topology on a set expressed as a topological space, using implicit structure indices. The advantage of this version over indistpsx 20814 is that it is independent of the indices of the component definitions df-base 15863 and df-tset 15960, and if they are changed in the future, this theorem will not be affected. The advantage over indistps2 20816 is that it is easy to eliminate the hypotheses with eqid 2622 and vtoclg 3266 to result in a closed theorem. Theorems indistpsALT 20817 and indistps2ALT 20818 show that the two forms can be derived from each other. (Contributed by FL, 19-Jul-2006.) |
TopSet | ||
Theorem | indistps2 20816 | The indiscrete topology on a set expressed as a topological space, using direct component assignments. Compare with indistps 20815. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 20817 and indistps2ALT 20818 show that the two forms can be derived from each other. (Contributed by NM, 24-Oct-2012.) |
Theorem | indistpsALT 20817 | The indiscrete topology on a set expressed as a topological space. Here we show how to derive the structural version indistps 20815 from the direct component assignment version indistps2 20816. (Contributed by NM, 24-Oct-2012.) (New usage is discouraged.) (Proof modification is discouraged.) |
TopSet | ||
Theorem | indistps2ALT 20818 | The indiscrete topology on a set expressed as a topological space, using direct component assignments. Here we show how to derive the direct component assignment version indistps2 20816 from the structural version indistps 20815. (Contributed by NM, 24-Oct-2012.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | distps 20819 | The discrete topology on a set expressed as a topological space. (Contributed by FL, 20-Aug-2006.) |
TopSet | ||
Syntax | ccld 20820 | Extend class notation with the set of closed sets of a topology. |
Syntax | cnt 20821 | Extend class notation with interior of a subset of a topology base set. |
Syntax | ccl 20822 | Extend class notation with closure of a subset of a topology base set. |
Definition | df-cld 20823* | Define a function on topologies whose value is the set of closed sets of the topology. (Contributed by NM, 2-Oct-2006.) |
Definition | df-ntr 20824* | Define a function on topologies whose value is the interior function on the subsets of the base set. See ntrval 20840. (Contributed by NM, 10-Sep-2006.) |
Definition | df-cls 20825* | Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 20841. (Contributed by NM, 3-Oct-2006.) |
Theorem | fncld 20826 | The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Theorem | cldval 20827* | The set of closed sets of a topology. (Note that the set of open sets is just the topology itself, so we don't have a separate definition.) (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Theorem | ntrfval 20828* | The interior function on the subsets of a topology's base set. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Theorem | clsfval 20829* | The closure function on the subsets of a topology's base set. (Contributed by NM, 3-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Theorem | cldrcl 20830 | Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Theorem | iscld 20831 | The predicate " is a closed set." (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Theorem | iscld2 20832 | A subset of the underlying set of a topology is closed iff its complement is open. (Contributed by NM, 4-Oct-2006.) |
Theorem | cldss 20833 | A closed set is a subset of the underlying set of a topology. (Contributed by NM, 5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.) |
Theorem | cldss2 20834 | The set of closed sets is contained in the powerset of the base. (Contributed by Mario Carneiro, 6-Jan-2014.) |
Theorem | cldopn 20835 | The complement of a closed set is open. (Contributed by NM, 5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.) |
Theorem | isopn2 20836 | A subset of the underlying set of a topology is open iff its complement is closed. (Contributed by NM, 4-Oct-2006.) |
Theorem | opncld 20837 | The complement of an open set is closed. (Contributed by NM, 6-Oct-2006.) |
Theorem | difopn 20838 | The difference of a closed set with an open set is open. (Contributed by Mario Carneiro, 6-Jan-2014.) |
Theorem | topcld 20839 | The underlying set of a topology is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 3-Oct-2006.) |
Theorem | ntrval 20840 | The interior of a subset of a topology's base set is the union of all the open sets it includes. Definition of interior of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Theorem | clsval 20841* | The closure of a subset of a topology's base set is the intersection of all the closed sets that include it. Definition of closure of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Theorem | 0cld 20842 | The empty set is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 4-Oct-2006.) |
Theorem | iincld 20843* | The indexed intersection of a collection of closed sets is closed. Theorem 6.1(2) of [Munkres] p. 93. (Contributed by NM, 5-Oct-2006.) (Revised by Mario Carneiro, 3-Sep-2015.) |
Theorem | intcld 20844 | The intersection of a set of closed sets is closed. (Contributed by NM, 5-Oct-2006.) |
Theorem | uncld 20845 | The union of two closed sets is closed. Equivalent to Theorem 6.1(3) of [Munkres] p. 93. (Contributed by NM, 5-Oct-2006.) |
Theorem | cldcls 20846 | A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.) |
Theorem | incld 20847 | The intersection of two closed sets is closed. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.) |
Theorem | riincld 20848* | An indexed relative intersection of closed sets is closed. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Theorem | iuncld 20849* | A finite indexed union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.) |
Theorem | unicld 20850 | A finite union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.) |
Theorem | clscld 20851 | The closure of a subset of a topology's underlying set is closed. (Contributed by NM, 4-Oct-2006.) |
Theorem | clsf 20852 | The closure function is a function from subsets of the base to closed sets. (Contributed by Mario Carneiro, 11-Apr-2015.) |
Theorem | ntropn 20853 | The interior of a subset of a topology's underlying set is open. (Contributed by NM, 11-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Theorem | clsval2 20854 | Express closure in terms of interior. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Theorem | ntrval2 20855 | Interior expressed in terms of closure. (Contributed by NM, 1-Oct-2007.) |
Theorem | ntrdif 20856 | An interior of a complement is the complement of the closure. This set is also known as the exterior of . (Contributed by Jeff Hankins, 31-Aug-2009.) |
Theorem | clsdif 20857 | A closure of a complement is the complement of the interior. (Contributed by Jeff Hankins, 31-Aug-2009.) |
Theorem | clsss 20858 | Subset relationship for closure. (Contributed by NM, 10-Feb-2007.) |
Theorem | ntrss 20859 | Subset relationship for interior. (Contributed by NM, 3-Oct-2007.) |
Theorem | sscls 20860 | A subset of a topology's underlying set is included in its closure. (Contributed by NM, 22-Feb-2007.) |
Theorem | ntrss2 20861 | A subset includes its interior. (Contributed by NM, 3-Oct-2007.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Theorem | ssntr 20862 | An open subset of a set is a subset of the set's interior. (Contributed by Jeff Hankins, 31-Aug-2009.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Theorem | clsss3 20863 | The closure of a subset of a topological space is included in the space. (Contributed by NM, 26-Feb-2007.) |
Theorem | ntrss3 20864 | The interior of a subset of a topological space is included in the space. (Contributed by NM, 1-Oct-2007.) |
Theorem | ntrin 20865 | A pairwise intersection of interiors is the interior of the intersection. This does not always hold for arbitrary intersections. (Contributed by Jeff Hankins, 31-Aug-2009.) |
Theorem | cmclsopn 20866 | The complement of a closure is open. (Contributed by NM, 11-Sep-2006.) |
Theorem | cmntrcld 20867 | The complement of an interior is closed. (Contributed by NM, 1-Oct-2007.) (Proof shortened by OpenAI, 3-Jul-2020.) |
Theorem | iscld3 20868 | A subset is closed iff it equals its own closure. (Contributed by NM, 2-Oct-2006.) |
Theorem | iscld4 20869 | A subset is closed iff it contains its own closure. (Contributed by NM, 31-Jan-2008.) |
Theorem | isopn3 20870 | A subset is open iff it equals its own interior. (Contributed by NM, 9-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Theorem | clsidm 20871 | The closure operation is idempotent. (Contributed by NM, 2-Oct-2007.) |
Theorem | ntridm 20872 | The interior operation is idempotent. (Contributed by NM, 2-Oct-2007.) |
Theorem | clstop 20873 | The closure of a topology's underlying set is entire set. (Contributed by NM, 5-Oct-2007.) |
Theorem | ntrtop 20874 | The interior of a topology's underlying set is entire set. (Contributed by NM, 12-Sep-2006.) |
Theorem | 0ntr 20875 | A subset with an empty interior cannot cover a whole (nonempty) topology. (Contributed by NM, 12-Sep-2006.) |
Theorem | clsss2 20876 | If a subset is included in a closed set, so is the subset's closure. (Contributed by NM, 22-Feb-2007.) |
Theorem | elcls 20877* | Membership in a closure. Theorem 6.5(a) of [Munkres] p. 95. (Contributed by NM, 22-Feb-2007.) |
Theorem | elcls2 20878* | Membership in a closure. (Contributed by NM, 5-Mar-2007.) |
Theorem | clsndisj 20879 | Any open set containing a point that belongs to the closure of a subset intersects the subset. One direction of Theorem 6.5(a) of [Munkres] p. 95. (Contributed by NM, 26-Feb-2007.) |
Theorem | ntrcls0 20880 | A subset whose closure has an empty interior also has an empty interior. (Contributed by NM, 4-Oct-2007.) |
Theorem | ntreq0 20881* | Two ways to say that a subset has an empty interior. (Contributed by NM, 3-Oct-2007.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Theorem | cldmre 20882 | The closed sets of a topology comprise a Moore system on the points of the topology. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
Moore | ||
Theorem | mrccls 20883 | Moore closure generalizes closure in a topology. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
mrCls | ||
Theorem | cls0 20884 | The closure of the empty set. (Contributed by NM, 2-Oct-2007.) |
Theorem | ntr0 20885 | The interior of the empty set. (Contributed by NM, 2-Oct-2007.) |
Theorem | isopn3i 20886 | An open subset equals its own interior. (Contributed by Mario Carneiro, 30-Dec-2016.) |
Theorem | elcls3 20887* | Membership in a closure in terms of the members of a basis. Theorem 6.5(b) of [Munkres] p. 95. (Contributed by NM, 26-Feb-2007.) (Revised by Mario Carneiro, 3-Sep-2015.) |
Theorem | opncldf1 20888* | A bijection useful for converting statements about open sets to statements about closed sets and vice versa. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.) |
Theorem | opncldf2 20889* | The values of the open-closed bijection. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.) |
Theorem | opncldf3 20890* | The values of the converse/inverse of the open-closed bijection. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.) |
Theorem | isclo 20891* | A set is clopen iff for every point in the space there is a neighborhood such that all the points in are in iff is. (Contributed by Mario Carneiro, 10-Mar-2015.) |
Theorem | isclo2 20892* | A set is clopen iff for every point in the space there is a neighborhood of which is either disjoint from or contained in . (Contributed by Mario Carneiro, 7-Jul-2015.) |
Theorem | discld 20893 | The open sets of a discrete topology are closed and its closed sets are open. (Contributed by FL, 7-Jun-2007.) (Revised by Mario Carneiro, 7-Apr-2015.) |
Theorem | sn0cld 20894 | The closed sets of the topology . (Contributed by FL, 5-Jan-2009.) |
Theorem | indiscld 20895 | The closed sets of an indiscrete topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 14-Aug-2015.) |
Theorem | mretopd 20896* | A Moore collection which is closed under finite unions called topological; such a collection is the closed sets of a canonically associated topology. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Moore TopOn | ||
Theorem | toponmre 20897 | The topologies over a given base set form a Moore collection: the intersection of any family of them is a topology, including the empty (relative) intersection which gives the discrete topology distop 20799. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
TopOn Moore | ||
Theorem | cldmreon 20898 | The closed sets of a topology over a set are a Moore collection over the same set. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
TopOn Moore | ||
Theorem | iscldtop 20899* | A family is the closed sets of a topology iff it is a Moore collection and closed under finite union. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
TopOn Moore | ||
Theorem | mreclatdemoBAD 20900 | The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclatBAD 17187. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 6611 update): This proof uses the old df-clat 17108 and references the required instance of mreclatBAD 17187 as a hypothesis. When mreclatBAD 17187 is corrected to become mreclat, delete this theorem and uncomment the mreclatdemo below. |
Moore toInc toInc |
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