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

Theoremllyrest 17501 An open subspace of a locally space is also locally . (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally t Locally

Theoremnllyrest 17502 An open subspace of an n-locally space is also n-locally . (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally t 𝑛Locally

Theoremloclly 17503 If is a local property, then both Locally and 𝑛Locally simplify to . (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally 𝑛Locally

Theoremllyidm 17504 Idempotence of the "locally" predicate, i.e. being "locally " is a local property. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally Locally Locally

Theoremnllyidm 17505 Idempotence of the "n-locally" predicate, i.e. being "n-locally " is a local property. (Use loclly 17503 to show 𝑛Locally 𝑛Locally 𝑛Locally .) (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally 𝑛Locally 𝑛Locally

Theoremtoplly 17506 A topology is locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally

Theoremtopnlly 17507 A topology is n-locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally

Theoremhauslly 17508 A Hausdorff space is locally Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally

Theoremhausnlly 17509 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 17510 A compact Hausdorff space is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally

Theoremcldllycmp 17511 A closed subspace of a locally compact space is also locally compact. (The analogous result for open subspaces follows from the more general nllyrest 17502.) (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally t 𝑛Locally

Theoremlly1stc 17512 First-countability is a local property (unlike second-countability). (Contributed by Mario Carneiro, 21-Mar-2015.)
Locally

Theoremdislly 17513* The discrete space is locally if and only if every singleton space has property . (Contributed by Mario Carneiro, 20-Mar-2015.)
Locally

Theoremdisllycmp 17514 A discrete space is locally compact. (Contributed by Mario Carneiro, 20-Mar-2015.)
Locally

Theoremdis1stc 17515 A discrete space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremhausmapdom 17516 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 17973 to see a weaker result if the assumption of first-countability is omitted.) (Contributed by Mario Carneiro, 9-Apr-2015.)

Theoremhauspwdom 17517 Simplify the cardinal of hausmapdom 17516 to when is an infinite cardinal greater than . (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)

11.1.15  Compactly generated spaces

Syntaxckgen 17518 Extend class notation with the compact generator operation.
𝑘Gen

Definitiondf-kgen 17519* 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 17520* 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 17521* Value of the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)
TopOn 𝑘Gen t t

Theoremkgeni 17522 Property of the open sets in the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen t t

Theoremkgentopon 17523 The compact generator generates a topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
TopOn 𝑘Gen TopOn

Theoremkgenuni 17524 The base set of the compact generator is the same as the original topology. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen

Theoremkgenftop 17525 The compact generator generates a topology. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen

Theoremkgenf 17526 The compact generator is a function on topologies. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen

Theoremkgentop 17527 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 17528 The compact generator generates a finer topology than the original. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen

Theoremkgenhaus 17529 The compact generator generates another Hausdorff topology given a Hausdorff topology to start from. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑘Gen

Theoremkgencmp 17530 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 17531 The compact generator topology has the same compact sets as the original topology. (Contributed by Mario Carneiro, 20-Mar-2015.)
t 𝑘Gent

Theoremkgenidm 17532 The compact generator is idempotent on compactly generated spaces. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen 𝑘Gen

Theoremiskgen2 17533 A space is compactly generated iff it contains its image under the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen 𝑘Gen

Theoremiskgen3 17534* 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 17535* A locally compact space is compactly generated. (This variant of llycmpkgen 17537 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 17536 A compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑘Gen

Theoremllycmpkgen 17537 A locally compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑛Locally 𝑘Gen

Theorem1stckgenlem 17538 The one-point compactification of is compact. (Contributed by Mario Carneiro, 21-Mar-2015.)
TopOn                     t

Theorem1stckgen 17539 A first-countable space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑘Gen

Theoremkgen2ss 17540 The compact generator preserves the subset (fineness) relationship on topologies. (Contributed by Mario Carneiro, 21-Mar-2015.)
TopOn TopOn 𝑘Gen 𝑘Gen

Theoremkgencn 17541* 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 17542* 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 17543 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 17544 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

11.1.16  Product topologies

Syntaxctx 17545 Extend class notation with the binary topological product operation.

Syntaxcxko 17546 Extend class notation with a function whose value is the compact-open topology.

Definitiondf-tx 17547* 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 17548* 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 17549* Value of the binary topological product operation. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 30-Aug-2015.)

Theoremtxuni2 17550* The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremtxbasex 17551* The basis for the product topology is a set. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtxbas 17552* 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.)

Theoremeltx 17553* A set in a product is open iff each point is surrounded by an open rectangle. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Theoremtxtop 17554 The product of two topologies is a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremptval 17555* The value of the product topology function. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremptpjpre1 17556* The preimage of a projection function can be expressed as an indexed cartesian product. (Contributed by Mario Carneiro, 6-Feb-2015.)

Theoremelpt 17557* Elementhood in the bases of a product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremelptr 17558* A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremelptr2 17559* A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremptbasid 17560* The base set of the product topology is a basic open set. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremptuni2 17561* The base set for the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremptbasin 17562* The basis for a product topology is closed under intersections. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremptbasin2 17563* The basis for a product topology is closed under intersections. (Contributed by Mario Carneiro, 19-Mar-2015.)

Theoremptbas 17564* The basis for a product topology is a basis. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremptpjpre2 17565* The basis for a product topology is a basis. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremptbasfi 17566* 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.)

Theorempttop 17567 The product topology is a topology. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremptopn 17568* A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremptopn2 17569* A sub-basic open set in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremxkotf 17570* Functionality of function . (Contributed by Mario Carneiro, 19-Mar-2015.)
t

Theoremxkobval 17571* Alternative expression for the subbase of the compact-open topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
t               t

Theoremxkoval 17572* Value of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
t

Theoremxkotop 17573 The compact-open topology is a topology. (Contributed by Mario Carneiro, 19-Mar-2015.)

Theoremxkoopn 17574* A basic open set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
t

Theoremtxtopi 17575 The product of two topologies is a topology. (Contributed by Jeff Madsen, 15-Jun-2010.)

Theoremtxtopon 17576 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

Theoremtxuni 17577 The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremtxunii 17578 The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 15-Jun-2010.)

Theoremptuni 17579* The base set for the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremptunimpt 17580* Base set of a product topology given by substitution. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theorempttopon 17581* The base set for the product topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
TopOn TopOn

Theorempttoponconst 17582 The base set for a product topology when all factors are the same. (Contributed by Mario Carneiro, 22-Aug-2015.)
TopOn TopOn

Theoremptuniconst 17583 The base set for a product topology when all factors are the same. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremxkouni 17584 The base set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)

Theoremxkotopon 17585 The base set of the compact-open topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
TopOn

Theoremptval2 17586* The value of the product topology function. (Contributed by Mario Carneiro, 7-Feb-2015.)

Theoremtxopn 17587 The product of two open sets is open in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremtxcld 17588 The product of two closed sets is closed in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)

Theoremtxcls 17589 Closure of a rectangle in the product topology. (Contributed by Mario Carneiro, 17-Sep-2015.)
TopOn TopOn

Theoremtxss12 17590 Subset property of the topological product. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtxbasval 17591 It is sufficient to consider products of the bases for the topologies in the topological product. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremneitx 17592 The cross product of two neighborhoods is a neighborhood in the product topology. (Contributed by Thierry Arnoux, 13-Jan-2018.)

Theoremtxcnpi 17593* Continuity of a two-argument function at a point. (Contributed by Mario Carneiro, 20-Sep-2015.)
TopOn       TopOn

Theoremtx1cn 17594 Continuity of the first projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
TopOn TopOn

Theoremtx2cn 17595 Continuity of the second projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
TopOn TopOn

Theoremptpjcn 17596* Continuity of a projection map into a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Feb-2015.)

Theoremptpjopn 17597* The projection map is an open map. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremptcld 17598* A closed box in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremptcldmpt 17599* A closed box in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremptclsg 17600* The closure of a box in the product topology is the box formed from the closures of the factors. The proof uses the axiom of choice; the last hypothesis is the choice assumption. (Contributed by Mario Carneiro, 3-Sep-2015.)
TopOn              AC

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