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Theorem List for Metamath Proof Explorer - 17501-17600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremt1conperf 17501 A connected T1 space is perfect, unless it is the topology of a singleton. (Contributed by Mario Carneiro, 26-Dec-2016.)
Perf

11.1.14  First- and second-countability

Syntaxc1stc 17502 Extend class definition to include the class of all first-countable topologies.

Syntaxc2ndc 17503 Extend class definition to include the class of all second-countable topologies.

Definitiondf-1stc 17504* Define the class of all first-countable topologies. (Contributed by Jeff Hankins, 22-Aug-2009.)

Definitiondf-2ndc 17505* Define the class of all second-countable topologies. (Contributed by Jeff Hankins, 17-Jan-2010.)

Theoremis1stc 17506* The predicate "is a first-countable topology." This can be described as "every point has a countable local basis" - that is, every point has a countable collection of open sets containing it such that every open set containing the point has an open set from this collection as a subset. (Contributed by Jeff Hankins, 22-Aug-2009.)

Theoremis1stc2 17507* An equivalent way of saying "is a first-countable topology." (Contributed by Jeff Hankins, 22-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)

Theorem1stctop 17508 A first-countable topology is a topology. (Contributed by Jeff Hankins, 22-Aug-2009.)

Theorem1stcclb 17509* A property of points in a first-countable topology. (Contributed by Jeff Hankins, 22-Aug-2009.)

Theorem1stcfb 17510* For any point in a first-countable topology, there is a function enumerating neighborhoods of which is decreasing and forms a local base. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremis2ndc 17511* The property of being second-countable. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.)

Theorem2ndctop 17512 A second-countable topology is a topology. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.)

Theorem2ndci 17513 A countable basis generates a second-countable topology. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theorem2ndcsb 17514* Having a countable subbase is a sufficient condition for second-countability. (Contributed by Jeff Hankins, 17-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Theorem2ndcredom 17515 A second-countable space has at most the cardinality of the continuum. (Contributed by Mario Carneiro, 9-Apr-2015.)

Theorem2ndc1stc 17516 A second-countable space is first-countable. (Contributed by Jeff Hankins, 17-Jan-2010.)

Theorem1stcrestlem 17517* Lemma for 1stcrest 17518. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theorem1stcrest 17518 A subspace of a first-countable space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
t

Theorem2ndcrest 17519 A subspace of a second-countable space is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
t

Theorem2ndcctbss 17520* If a topology is second-countable, every base has a countable subset which is a base. Exercise 16B2 in Willard. (Contributed by Jeff Hankins, 28-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Theorem2ndcdisj 17521* Any disjoint family of open sets in a second-countable space is countable. (The sets are required to be nonempty because otherwise there could be many empty sets in the family.) (Contributed by Mario Carneiro, 21-Mar-2015.) (Proof shortened by Mario Carneiro, 9-Apr-2015.) (Revised by NM, 17-Jun-2017.)

Theorem2ndcdisj2 17522* Any disjoint collection of open sets in a second-countable space is countable. (Contributed by Mario Carneiro, 21-Mar-2015.) (Proof shortened by Mario Carneiro, 9-Apr-2015.) (Revised by NM, 17-Jun-2017.)

Theorem2ndcomap 17523* A surjective continuous open map maps second-countable spaces to second-countable spaces. (Contributed by Mario Carneiro, 9-Apr-2015.)

Theorem2ndcsep 17524* A second-countable topology is separable, which is to say it contains a countable dense subset. (Contributed by Mario Carneiro, 13-Apr-2015.)

Theoremdis2ndc 17525 A discrete space is second-countable iff it is countable. (Contributed by Mario Carneiro, 13-Apr-2015.)

Theorem1stcelcls 17526* A point belongs to the closure of a subset iff there is a sequence in the subset converging to it. Theorem 1.4-6(a) of [Kreyszig] p. 30. This proof uses countable choice ax-cc 8317. A space satisfying the conclusion of this theorem is called a sequential space, so the theorem can also be stated as "every first-countable space is a sequential space". (Contributed by Mario Carneiro, 21-Mar-2015.)

Theorem1stccnp 17527* A mapping is continuous at in a first-countable space iff it is sequentially continuous at , meaning that the image under of every sequence converging at converges to . This proof uses ax-cc 8317, but only via 1stcelcls 17526, so it could be refactored into a proof that continuity and sequential continuity are the same in sequential spaces. (Contributed by Mario Carneiro, 7-Sep-2015.)
TopOn       TopOn

Theorem1stccn 17528* A mapping , where is first-countable, is continuous iff it is sequentially continuous, meaning that for any sequence converging to , its image under converges to . (Contributed by Mario Carneiro, 7-Sep-2015.)
TopOn       TopOn

11.1.15  Local topological properties

Syntaxclly 17529 Extend class notation with the "locally " predicate of a topological space.
Locally

Syntaxcnlly 17530 Extend class notation with the "N-locally " predicate of a topological space.
𝑛Locally

Definitiondf-lly 17531* Define a space that is locally , where is a topological property like "compact", "connected", or "path-connected". A topological space is locally if every neighborhood of a point contains an open sub-neighborhood that is in the subspace topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally t

Definitiondf-nlly 17532* Define a space that is n-locally , where is a topological property like "compact", "connected", or "path-connected". A topological space is n-locally if every neighborhood of a point contains a sub-neighborhood that is in the subspace topology.

The terminology "n-locally", where 'n' stands for "neighborhood", is not standard, although this is sometimes called "weakly locally ". The reason for the distinction is that some notions only make sense for arbitrary neighborhoods (such as "locally compact", which is actually 𝑛Locally in our teminology - open compact sets are not very useful), while others such as "locally connected" are strictly weaker notions if the neighborhoods are not required to be open. (Contributed by Mario Carneiro, 2-Mar-2015.)

𝑛Locally t

Theoremislly 17533* The property of being a locally topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally t

Theoremisnlly 17534* The property of being an n-locally topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally t

Theoremllyeq 17535 Equality theorem for the Locally predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally Locally

Theoremnllyeq 17536 Equality theorem for the Locally predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally 𝑛Locally

Theoremllytop 17537 A locally space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally

Theoremnllytop 17538 A locally space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally

Theoremllyi 17539* The property of a locally topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally t

Theoremnllyi 17540* The property of an n-locally topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally t

Theoremnlly2i 17541* Eliminate the neighborhood symbol from nllyi 17540. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally t

Theoremllynlly 17542 A locally space is n-locally : the "n-locally" predicate is the weaker notion. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally 𝑛Locally

Theoremllyssnlly 17543 A locally space is n-locally . (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally 𝑛Locally

Theoremllyss 17544 The "locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally Locally

Theoremnllyss 17545 The "n-locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally 𝑛Locally

Theoremsubislly 17546* The property of a subspace being locally . (Contributed by Mario Carneiro, 10-Mar-2015.)
t Locally t

Theoremrestnlly 17547* If the property passes to open subspaces, then a space is n-locally iff it is locally . (Contributed by Mario Carneiro, 2-Mar-2015.)
t        𝑛Locally Locally

Theoremrestlly 17548* If the property passes to open subspaces, then a space which is is also locally . (Contributed by Mario Carneiro, 2-Mar-2015.)
t               Locally

Theoremislly2 17549* An alternative expression for Locally when passes to open subspaces: A space is locally if every point is contained in an open neighborhood with property . (Contributed by Mario Carneiro, 2-Mar-2015.)
t               Locally t

Theoremllyrest 17550 An open subspace of a locally space is also locally . (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally t Locally

Theoremnllyrest 17551 An open subspace of an n-locally space is also n-locally . (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally t 𝑛Locally

Theoremloclly 17552 If is a local property, then both Locally and 𝑛Locally simplify to . (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally 𝑛Locally

Theoremllyidm 17553 Idempotence of the "locally" predicate, i.e. being "locally " is a local property. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally Locally Locally

Theoremnllyidm 17554 Idempotence of the "n-locally" predicate, i.e. being "n-locally " is a local property. (Use loclly 17552 to show 𝑛Locally 𝑛Locally 𝑛Locally .) (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally 𝑛Locally 𝑛Locally

Theoremtoplly 17555 A topology is locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally

Theoremtopnlly 17556 A topology is n-locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally

Theoremhauslly 17557 A Hausdorff space is locally Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally

Theoremhausnlly 17558 A Hausdorff space is n-locally Hausdorff iff it is locally Hausdorff (both notions are thus referred to as "locally Hausdorff"). (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally Locally

Theoremhausllycmp 17559 A compact Hausdorff space is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally

Theoremcldllycmp 17560 A closed subspace of a locally compact space is also locally compact. (The analogous result for open subspaces follows from the more general nllyrest 17551.) (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally t 𝑛Locally

Theoremlly1stc 17561 First-countability is a local property (unlike second-countability). (Contributed by Mario Carneiro, 21-Mar-2015.)
Locally

Theoremdislly 17562* The discrete space is locally if and only if every singleton space has property . (Contributed by Mario Carneiro, 20-Mar-2015.)
Locally

Theoremdisllycmp 17563 A discrete space is locally compact. (Contributed by Mario Carneiro, 20-Mar-2015.)
Locally

Theoremdis1stc 17564 A discrete space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremhausmapdom 17565 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 18022 to see a weaker result if the assumption of first-countability is omitted.) (Contributed by Mario Carneiro, 9-Apr-2015.)

Theoremhauspwdom 17566 Simplify the cardinal of hausmapdom 17565 to when is an infinite cardinal greater than . (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)

11.1.16  Compactly generated spaces

Syntaxckgen 17567 Extend class notation with the compact generator operation.
𝑘Gen

Definitiondf-kgen 17568* 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

Theoremkgenval 17569* 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

Theoremelkgen 17570* Value of the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)
TopOn 𝑘Gen t t

Theoremkgeni 17571 Property of the open sets in the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen t t

Theoremkgentopon 17572 The compact generator generates a topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
TopOn 𝑘Gen TopOn

Theoremkgenuni 17573 The base set of the compact generator is the same as the original topology. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen

Theoremkgenftop 17574 The compact generator generates a topology. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen

Theoremkgenf 17575 The compact generator is a function on topologies. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen

Theoremkgentop 17576 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

Theoremkgenss 17577 The compact generator generates a finer topology than the original. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen

Theoremkgenhaus 17578 The compact generator generates another Hausdorff topology given a Hausdorff topology to start from. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑘Gen

Theoremkgencmp 17579 The compact generator topology is the same as the original topology on compact subspaces. (Contributed by Mario Carneiro, 20-Mar-2015.)
t t 𝑘Gent

Theoremkgencmp2 17580 The compact generator topology has the same compact sets as the original topology. (Contributed by Mario Carneiro, 20-Mar-2015.)
t 𝑘Gent

Theoremkgenidm 17581 The compact generator is idempotent on compactly generated spaces. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen 𝑘Gen

Theoremiskgen2 17582 A space is compactly generated iff it contains its image under the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen 𝑘Gen

Theoremiskgen3 17583* 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

Theoremllycmpkgen2 17584* A locally compact space is compactly generated. (This variant of llycmpkgen 17586 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

Theoremcmpkgen 17585 A compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑘Gen

Theoremllycmpkgen 17586 A locally compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑛Locally 𝑘Gen

Theorem1stckgenlem 17587 The one-point compactification of is compact. (Contributed by Mario Carneiro, 21-Mar-2015.)
TopOn                     t

Theorem1stckgen 17588 A first-countable space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑘Gen

Theoremkgen2ss 17589 The compact generator preserves the subset (fineness) relationship on topologies. (Contributed by Mario Carneiro, 21-Mar-2015.)
TopOn TopOn 𝑘Gen 𝑘Gen

Theoremkgencn 17590* 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

Theoremkgencn2 17591* 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

Theoremkgencn3 17592 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

Theoremkgen2cn 17593 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

11.1.17  Product topologies

Syntaxctx 17594 Extend class notation with the binary topological product operation.

Syntaxcxko 17595 Extend class notation with a function whose value is the compact-open topology.

Definitiondf-tx 17596* 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.)

Definitiondf-xko 17597* 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

Theoremtxval 17598* Value of the binary topological product operation. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 30-Aug-2015.)

Theoremtxuni2 17599* The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremtxbasex 17600* The basis for the product topology is a set. (Contributed by Mario Carneiro, 2-Sep-2015.)

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