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

Definitiondf-kgen 17501* 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 17502* 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 17503* Value of the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)
TopOn 𝑘Gen t t

Theoremkgeni 17504 Property of the open sets in the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen t t

Theoremkgentopon 17505 The compact generator generates a topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
TopOn 𝑘Gen TopOn

Theoremkgenuni 17506 The base set of the compact generator is the same as the original topology. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen

Theoremkgenftop 17507 The compact generator generates a topology. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen

Theoremkgenf 17508 The compact generator is a function on topologies. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen

Theoremkgentop 17509 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 17510 The compact generator generates a finer topology than the original. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen

Theoremkgenhaus 17511 The compact generator generates another Hausdorff topology given a Hausdorff topology to start from. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑘Gen

Theoremkgencmp 17512 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 17513 The compact generator topology has the same compact sets as the original topology. (Contributed by Mario Carneiro, 20-Mar-2015.)
t 𝑘Gent

Theoremkgenidm 17514 The compact generator is idempotent on compactly generated spaces. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen 𝑘Gen

Theoremiskgen2 17515 A space is compactly generated iff it contains its image under the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen 𝑘Gen

Theoremiskgen3 17516* 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 17517* A locally compact space is compactly generated. (This variant of llycmpkgen 17519 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 17518 A compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑘Gen

Theoremllycmpkgen 17519 A locally compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑛Locally 𝑘Gen

Theorem1stckgenlem 17520 The one-point compactification of is compact. (Contributed by Mario Carneiro, 21-Mar-2015.)
TopOn                     t

Theorem1stckgen 17521 A first-countable space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑘Gen

Theoremkgen2ss 17522 The compact generator preserves the subset (fineness) relationship on topologies. (Contributed by Mario Carneiro, 21-Mar-2015.)
TopOn TopOn 𝑘Gen 𝑘Gen

Theoremkgencn 17523* 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 17524* 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 17525 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 17526 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 17527 Extend class notation with the binary topological product operation.

Syntaxcxko 17528 Extend class notation with a function whose value is the compact-open topology.

Definitiondf-tx 17529* 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 17530* 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 17531* Value of the binary topological product operation. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 30-Aug-2015.)

Theoremtxuni2 17532* The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremtxbasex 17533* The basis for the product topology is a set. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtxbas 17534* 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 17535* 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 17536 The product of two topologies is a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremptval 17537* The value of the product topology function. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremptpjpre1 17538* The preimage of a projection function can be expressed as an indexed cartesian product. (Contributed by Mario Carneiro, 6-Feb-2015.)

Theoremelpt 17539* Elementhood in the bases of a product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremelptr 17540* A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremelptr2 17541* A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremptbasid 17542* The base set of the product topology is a basic open set. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremptuni2 17543* The base set for the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremptbasin 17544* The basis for a product topology is closed under intersections. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremptbasin2 17545* The basis for a product topology is closed under intersections. (Contributed by Mario Carneiro, 19-Mar-2015.)

Theoremptbas 17546* The basis for a product topology is a basis. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremptpjpre2 17547* The basis for a product topology is a basis. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremptbasfi 17548* 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 17549 The product topology is a topology. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremptopn 17550* A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremptopn2 17551* A sub-basic open set in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremxkotf 17552* Functionality of function . (Contributed by Mario Carneiro, 19-Mar-2015.)
t

Theoremxkobval 17553* Alternative expression for the subbase of the compact-open topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
t               t

Theoremxkoval 17554* Value of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
t

Theoremxkotop 17555 The compact-open topology is a topology. (Contributed by Mario Carneiro, 19-Mar-2015.)

Theoremxkoopn 17556* A basic open set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
t

Theoremtxtopi 17557 The product of two topologies is a topology. (Contributed by Jeff Madsen, 15-Jun-2010.)

Theoremtxtopon 17558 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 17559 The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremtxunii 17560 The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 15-Jun-2010.)

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

Theoremptunimpt 17562* Base set of a product topology given by substitution. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theorempttopon 17563* The base set for the product topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
TopOn TopOn

Theorempttoponconst 17564 The base set for a product topology when all factors are the same. (Contributed by Mario Carneiro, 22-Aug-2015.)
TopOn TopOn

Theoremptuniconst 17565 The base set for a product topology when all factors are the same. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremxkouni 17566 The base set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)

Theoremxkotopon 17567 The base set of the compact-open topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
TopOn

Theoremptval2 17568* The value of the product topology function. (Contributed by Mario Carneiro, 7-Feb-2015.)

Theoremtxopn 17569 The product of two open sets is open in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremtxcld 17570 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 17571 Closure of a rectangle in the product topology. (Contributed by Mario Carneiro, 17-Sep-2015.)
TopOn TopOn

Theoremtxss12 17572 Subset property of the topological product. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtxbasval 17573 It is sufficient to consider products of the bases for the topologies in the topological product. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremneitx 17574 The cross product of two neighborhoods is a neighborhood in the product topology. (Contributed by Thierry Arnoux, 13-Jan-2018.)

Theoremtxcnpi 17575* Continuity of a two-argument function at a point. (Contributed by Mario Carneiro, 20-Sep-2015.)
TopOn       TopOn

Theoremtx1cn 17576 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 17577 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 17578* Continuity of a projection map into a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Feb-2015.)

Theoremptpjopn 17579* The projection map is an open map. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremptcld 17580* A closed box in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremptcldmpt 17581* A closed box in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremptclsg 17582* 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

Theoremptcls 17583* The closure of a box in the product topology is the box formed from the closures of the factors. This theorem is an AC equivalent. (Contributed by Mario Carneiro, 2-Sep-2015.)
TopOn

Theoremdfac14lem 17584* Lemma for dfac14 17585. By equipping for some with the particular point topology, we can show that is in the closure of ; hence the sequence is in the product of the closures, and we can utilize this instance of ptcls 17583 to extract an element of the closure of . (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremdfac14 17585* Theorem ptcls 17583 is an equivalent of the axiom of choice. (Contributed by Mario Carneiro, 3-Sep-2015.)
CHOICE

Theoremxkoccn 17586* The "constant function" function which maps to the constant function is a continuous function from into the space of continuous functions from to . This can also be understood as the currying of the first projection function. (The currying of the second projection function is , which we already know is continuous because it is a constant function.) (Contributed by Mario Carneiro, 19-Mar-2015.)
TopOn TopOn

Theoremtxcnp 17587* If two functions are continuous at , then the ordered pair of them is continuous at into the product topology. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn       TopOn

Theoremptcnplem 17588* Lemma for ptcnp 17589. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn

Theoremptcnp 17589* If every projection of a function is continuous at , then the function itself is continuous at into the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn

Theoremupxp 17590* Universal property of the Cartesian product considered as a categorical product in the category of sets. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremtxcnmpt 17591* A map into the product of two topological spaces is continuous if both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremuptx 17592* Universal property of the binary topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)

Theoremtxcn 17593 A map into the product of two topological spaces is continuous iff both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)

Theoremptcn 17594* If every projection of a function is continuous, then the function itself is continuous into the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
TopOn

Theoremprdstopn 17595 Topology of a structure product. (Contributed by Mario Carneiro, 27-Aug-2015.)
s

Theoremprdstps 17596 A structure product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
s

Theorempwstps 17597 A structure product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
s

Theoremtxrest 17598 The subspace of a topological product space induced by a subset with a Cartesian product representation is a topological product of the subspaces induced by the subspaces of the terms of the products. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
t t t

Theoremtxdis 17599 The topological product of discrete spaces is discrete. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremtxindislem 17600 Lemma for txindis 17601. (Contributed by Mario Carneiro, 14-Aug-2015.)

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