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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | disllycmp 21301 | A discrete space is locally compact. (Contributed by Mario Carneiro, 20-Mar-2015.) |
Locally | ||
Theorem | dis1stc 21302 | A discrete space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Theorem | hausmapdom 21303 | If is a first-countable Hausdorff space, then the cardinality of the closure of a set is bounded by to the power . In particular, a first-countable Hausdorff space with a dense subset has cardinality at most , and a separable first-countable Hausdorff space has cardinality at most . (Compare hauspwpwdom 21792 to see a weaker result if the assumption of first-countability is omitted.) (Contributed by Mario Carneiro, 9-Apr-2015.) |
Theorem | hauspwdom 21304 | Simplify the cardinal of hausmapdom 21303 to when is an infinite cardinal greater than . (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Syntax | cref 21305 | Extend class definition to include the refinement relation. |
Syntax | cptfin 21306 | Extend class definition to include the class of point-finite covers. |
Syntax | clocfin 21307 | Extend class definition to include the class of locally finite covers. |
Definition | df-ref 21308* | Define the refinement relation. (Contributed by Jeff Hankins, 18-Jan-2010.) |
Definition | df-ptfin 21309* | Define "point-finite." (Contributed by Jeff Hankins, 21-Jan-2010.) |
Definition | df-locfin 21310* | Define "locally finite." (Contributed by Jeff Hankins, 21-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.) |
Theorem | refrel 21311 | Refinement is a relation. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.) |
Theorem | isref 21312* | The property of being a refinement of a cover. Dr. Nyikos once commented in class that the term "refinement" is actually misleading and that people are inclined to confuse it with the notion defined in isfne 32334. On the other hand, the two concepts do seem to have a dual relationship. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.) |
Theorem | refbas 21313 | A refinement covers the same set. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.) |
Theorem | refssex 21314* | Every set in a refinement has a superset in the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.) |
Theorem | ssref 21315 | A subcover is a refinement of the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.) |
Theorem | refref 21316 | Reflexivity of refinement. (Contributed by Jeff Hankins, 18-Jan-2010.) |
Theorem | reftr 21317 | Refinement is transitive. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.) |
Theorem | refun0 21318 | Adding the empty set preserves refinements. (Contributed by Thierry Arnoux, 31-Jan-2020.) |
Theorem | isptfin 21319* | The statement "is a point-finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.) |
Theorem | islocfin 21320* | The statement "is a locally finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.) |
Theorem | finptfin 21321 | A finite cover is a point-finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.) |
Theorem | ptfinfin 21322* | A point covered by a point-finite cover is only covered by finitely many elements. (Contributed by Jeff Hankins, 21-Jan-2010.) |
Theorem | finlocfin 21323 | A finite cover of a topological space is a locally finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.) |
Theorem | locfintop 21324 | A locally finite cover covers a topological space. (Contributed by Jeff Hankins, 21-Jan-2010.) |
Theorem | locfinbas 21325 | A locally finite cover must cover the base set of its corresponding topological space. (Contributed by Jeff Hankins, 21-Jan-2010.) |
Theorem | locfinnei 21326* | A point covered by a locally finite cover has a neighborhood which intersects only finitely many elements of the cover. (Contributed by Jeff Hankins, 21-Jan-2010.) |
Theorem | lfinpfin 21327 | A locally finite cover is point-finite. (Contributed by Jeff Hankins, 21-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.) |
Theorem | lfinun 21328 | Adding a finite set preserves locally finite covers. (Contributed by Thierry Arnoux, 31-Jan-2020.) |
Theorem | locfincmp 21329 | For a compact space, the locally finite covers are precisely the finite covers. Sadly, this property does not properly characterize all compact spaces. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.) |
Theorem | unisngl 21330* | Taking the union of the set of singletons recovers the initial set. (Contributed by Thierry Arnoux, 9-Jan-2020.) |
Theorem | dissnref 21331* | The set of singletons is a refinement of any open covering of the discrete topology. (Contributed by Thierry Arnoux, 9-Jan-2020.) |
Theorem | dissnlocfin 21332* | The set of singletons is locally finite in the discrete topology. (Contributed by Thierry Arnoux, 9-Jan-2020.) |
Theorem | locfindis 21333 | The locally finite covers of a discrete space are precisely the point-finite covers. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.) |
Theorem | locfincf 21334 | A locally finite cover in a coarser topology is locally finite in a finer topology. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.) |
TopOn | ||
Theorem | comppfsc 21335* | A space where every open cover has a point-finite subcover is compact. This is significant in part because it shows half of the proposition that if only half the generalization in the definition of metacompactness (and consequently paracompactness) is performed, one does not obtain any more spaces. (Contributed by Jeff Hankins, 21-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.) |
Syntax | ckgen 21336 | Extend class notation with the compact generator operation. |
𝑘Gen | ||
Definition | df-kgen 21337* | Define the "compact generator", the functor from topological spaces to compactly generated spaces. Also known as the k-ification operation. A set is k-open, i.e. 𝑘Gen, iff the preimage of is open in all compact Hausdorff spaces. (Contributed by Mario Carneiro, 20-Mar-2015.) |
𝑘Gen ↾t ↾t | ||
Theorem | kgenval 21338* | Value of the compact generator. (The "k" in 𝑘Gen comes from the name "k-space" for these spaces, after the German word kompakt.) (Contributed by Mario Carneiro, 20-Mar-2015.) |
TopOn 𝑘Gen ↾t ↾t | ||
Theorem | elkgen 21339* | Value of the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.) |
TopOn 𝑘Gen ↾t ↾t | ||
Theorem | kgeni 21340 | Property of the open sets in the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.) |
𝑘Gen ↾t ↾t | ||
Theorem | kgentopon 21341 | The compact generator generates a topology. (Contributed by Mario Carneiro, 22-Aug-2015.) |
TopOn 𝑘Gen TopOn | ||
Theorem | kgenuni 21342 | The base set of the compact generator is the same as the original topology. (Contributed by Mario Carneiro, 20-Mar-2015.) |
𝑘Gen | ||
Theorem | kgenftop 21343 | The compact generator generates a topology. (Contributed by Mario Carneiro, 20-Mar-2015.) |
𝑘Gen | ||
Theorem | kgenf 21344 | The compact generator is a function on topologies. (Contributed by Mario Carneiro, 20-Mar-2015.) |
𝑘Gen | ||
Theorem | kgentop 21345 | A compactly generated space is a topology. (Note: henceforth we will use the idiom " 𝑘Gen " to denote " is compactly generated", since as we will show a space is compactly generated iff it is in the range of the compact generator.) (Contributed by Mario Carneiro, 20-Mar-2015.) |
𝑘Gen | ||
Theorem | kgenss 21346 | The compact generator generates a finer topology than the original. (Contributed by Mario Carneiro, 20-Mar-2015.) |
𝑘Gen | ||
Theorem | kgenhaus 21347 | The compact generator generates another Hausdorff topology given a Hausdorff topology to start from. (Contributed by Mario Carneiro, 21-Mar-2015.) |
𝑘Gen | ||
Theorem | kgencmp 21348 | The compact generator topology is the same as the original topology on compact subspaces. (Contributed by Mario Carneiro, 20-Mar-2015.) |
↾t ↾t 𝑘Gen ↾t | ||
Theorem | kgencmp2 21349 | The compact generator topology has the same compact sets as the original topology. (Contributed by Mario Carneiro, 20-Mar-2015.) |
↾t 𝑘Gen ↾t | ||
Theorem | kgenidm 21350 | The compact generator is idempotent on compactly generated spaces. (Contributed by Mario Carneiro, 20-Mar-2015.) |
𝑘Gen 𝑘Gen | ||
Theorem | iskgen2 21351 | A space is compactly generated iff it contains its image under the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.) |
𝑘Gen 𝑘Gen | ||
Theorem | iskgen3 21352* | Derive the usual definition of "compactly generated". A topology is compactly generated if every subset of that is open in every compact subset is open. (Contributed by Mario Carneiro, 20-Mar-2015.) |
𝑘Gen ↾t ↾t | ||
Theorem | llycmpkgen2 21353* | A locally compact space is compactly generated. (This variant of llycmpkgen 21355 uses the weaker definition of locally compact, "every point has a compact neighborhood", instead of "every point has a local base of compact neighborhoods".) (Contributed by Mario Carneiro, 21-Mar-2015.) |
↾t 𝑘Gen | ||
Theorem | cmpkgen 21354 | A compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.) |
𝑘Gen | ||
Theorem | llycmpkgen 21355 | A locally compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.) |
𝑛Locally 𝑘Gen | ||
Theorem | 1stckgenlem 21356 | The one-point compactification of is compact. (Contributed by Mario Carneiro, 21-Mar-2015.) |
TopOn ↾t | ||
Theorem | 1stckgen 21357 | A first-countable space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.) |
𝑘Gen | ||
Theorem | kgen2ss 21358 | The compact generator preserves the subset (fineness) relationship on topologies. (Contributed by Mario Carneiro, 21-Mar-2015.) |
TopOn TopOn 𝑘Gen 𝑘Gen | ||
Theorem | kgencn 21359* | A function from a compactly generated space is continuous iff it is continuous "on compacta". (Contributed by Mario Carneiro, 21-Mar-2015.) |
TopOn TopOn 𝑘Gen ↾t ↾t | ||
Theorem | kgencn2 21360* | A function from a compactly generated space is continuous iff for all compact spaces and continuous , the composite is continuous. (Contributed by Mario Carneiro, 21-Mar-2015.) |
TopOn TopOn 𝑘Gen | ||
Theorem | kgencn3 21361 | The set of continuous functions from to is unaffected by k-ification of , if is already compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.) |
𝑘Gen 𝑘Gen | ||
Theorem | kgen2cn 21362 | A continuous function is also continuous with the domain and codomain replaced by their compact generator topologies. (Contributed by Mario Carneiro, 21-Mar-2015.) |
𝑘Gen 𝑘Gen | ||
Syntax | ctx 21363 | Extend class notation with the binary topological product operation. |
Syntax | cxko 21364 | Extend class notation with a function whose value is the compact-open topology. |
Definition | df-tx 21365* | Define the binary topological product, which is homeomorphic to the general topological product over a two element set, but is more convenient to use. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Definition | df-xko 21366* | Define the compact-open topology, which is the natural topology on the set of continuous functions between two topological spaces. (Contributed by Mario Carneiro, 19-Mar-2015.) |
↾t | ||
Theorem | txval 21367* | Value of the binary topological product operation. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 30-Aug-2015.) |
Theorem | txuni2 21368* | The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 31-Aug-2015.) |
Theorem | txbasex 21369* | The basis for the product topology is a set. (Contributed by Mario Carneiro, 2-Sep-2015.) |
Theorem | txbas 21370* | The set of Cartesian products of elements from two topological bases is a basis. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | eltx 21371* | A set in a product is open iff each point is surrounded by an open rectangle. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Theorem | txtop 21372 | The product of two topologies is a topology. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | ptval 21373* | The value of the product topology function. (Contributed by Mario Carneiro, 3-Feb-2015.) |
Theorem | ptpjpre1 21374* | The preimage of a projection function can be expressed as an indexed cartesian product. (Contributed by Mario Carneiro, 6-Feb-2015.) |
Theorem | elpt 21375* | Elementhood in the bases of a product topology. (Contributed by Mario Carneiro, 3-Feb-2015.) |
Theorem | elptr 21376* | A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.) |
Theorem | elptr2 21377* | A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.) |
Theorem | ptbasid 21378* | The base set of the product topology is a basic open set. (Contributed by Mario Carneiro, 3-Feb-2015.) |
Theorem | ptuni2 21379* | The base set for the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.) |
Theorem | ptbasin 21380* | The basis for a product topology is closed under intersections. (Contributed by Mario Carneiro, 3-Feb-2015.) |
Theorem | ptbasin2 21381* | The basis for a product topology is closed under intersections. (Contributed by Mario Carneiro, 19-Mar-2015.) |
Theorem | ptbas 21382* | The basis for a product topology is a basis. (Contributed by Mario Carneiro, 3-Feb-2015.) |
Theorem | ptpjpre2 21383* | The basis for a product topology is a basis. (Contributed by Mario Carneiro, 3-Feb-2015.) |
Theorem | ptbasfi 21384* | The basis for the product topology can also be written as the set of finite intersections of "cylinder sets", the preimages of projections into one factor from open sets in the factor. (We have to add itself to the list because if is empty we get while .) (Contributed by Mario Carneiro, 3-Feb-2015.) |
Theorem | pttop 21385 | The product topology is a topology. (Contributed by Mario Carneiro, 3-Feb-2015.) |
Theorem | ptopn 21386* | A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.) |
Theorem | ptopn2 21387* | A sub-basic open set in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Theorem | xkotf 21388* | Functionality of function . (Contributed by Mario Carneiro, 19-Mar-2015.) |
↾t | ||
Theorem | xkobval 21389* | Alternative expression for the subbase of the compact-open topology. (Contributed by Mario Carneiro, 23-Mar-2015.) |
↾t ↾t | ||
Theorem | xkoval 21390* | Value of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.) |
↾t | ||
Theorem | xkotop 21391 | The compact-open topology is a topology. (Contributed by Mario Carneiro, 19-Mar-2015.) |
Theorem | xkoopn 21392* | A basic open set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.) |
↾t | ||
Theorem | txtopi 21393 | The product of two topologies is a topology. (Contributed by Jeff Madsen, 15-Jun-2010.) |
Theorem | txtopon 21394 | The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 22-Aug-2015.) (Revised by Mario Carneiro, 2-Sep-2015.) |
TopOn TopOn TopOn | ||
Theorem | txuni 21395 | The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | txunii 21396 | The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 15-Jun-2010.) |
Theorem | ptuni 21397* | The base set for the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.) |
Theorem | ptunimpt 21398* | Base set of a product topology given by substitution. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Theorem | pttopon 21399* | The base set for the product topology. (Contributed by Mario Carneiro, 22-Aug-2015.) |
TopOn TopOn | ||
Theorem | pttoponconst 21400 | The base set for a product topology when all factors are the same. (Contributed by Mario Carneiro, 22-Aug-2015.) |
TopOn TopOn |
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