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Theorem List for Metamath Proof Explorer - 18001-18100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfclselbas 18001 A cluster point is in the base set. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)

Theoremfclsneii 18002 A neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfclssscls 18003 The set of cluster points is a subset of the closure of any filter element. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfclsnei 18004* Cluster points in terms of neighborhoods. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
TopOn

Theoremsupnfcls 18005* The filter of supersets of does not cluster at any point of the open set . (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)
TopOn

Theoremfclsbas 18006* Cluster points in terms of filter bases. (Contributed by Jeff Hankins, 13-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
TopOn

Theoremfclsss1 18007 A finer topology has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
TopOn

Theoremfclsss2 18008 A finer filter has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
TopOn

Theoremfclsrest 18009 The set of cluster points in a restricted topological space. (Contributed by Mario Carneiro, 15-Oct-2015.)
TopOn t t

Theoremfclscf 18010* Characterization of fineness of topologies in terms of cluster points. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
TopOn TopOn

Theoremflimfcls 18011 A limit point is a cluster point. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfclsfnflim 18012* A filter clusters at a point iff a finer filter converges to it. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)

Theoremflimfnfcls 18013* A filter converges to a point iff every finer filter clusters there. Along with fclsfnflim 18012, this theorem illustrates the duality between convergence and clustering. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfclscmpi 18014 Forward direction of fclscmp 18015. Every filter clusters in a compact space. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfclscmp 18015* A space is compact iff every filter clusters. (Contributed by Jeff Hankins, 20-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
TopOn

Theoremuffclsflim 18016 The cluster points of an ultrafilter are its limit points. (Contributed by Jeff Hankins, 11-Dec-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)

Theoremufilcmp 18017* A space is compact iff every ultrafilter converges. (Contributed by Jeff Hankins, 11-Dec-2009.) (Proof shortened by Mario Carneiro, 12-Apr-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)
UFL TopOn

Theoremfcfval 18018 The set of cluster points of a function. (Contributed by Jeff Hankins, 24-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
TopOn

Theoremisfcf 18019* The property of being a cluster point of a function. (Contributed by Jeff Hankins, 24-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
TopOn

Theoremfcfnei 18020* The property of being a cluster point of a function in terms of neighborhoods. (Contributed by Jeff Hankins, 26-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
TopOn

Theoremfcfelbas 18021 A cluster point of a function is in the base set of the topology. (Contributed by Jeff Hankins, 26-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
TopOn

Theoremfcfneii 18022 A neighborhood of a cluster point of a function contains a function value from every tail. (Contributed by Jeff Hankins, 27-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
TopOn

Theoremflfssfcf 18023 A limit point of a function is a cluster point of the function. (Contributed by Jeff Hankins, 28-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
TopOn

Theoremuffcfflf 18024 If the domain filter is an ultrafilter, the cluster points of the function are the limit points. (Contributed by Jeff Hankins, 12-Dec-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
TopOn

Theoremcnpfcfi 18025 Lemma for cnpfcf 18026. If a function is continuous at a point, it respects clustering there. (Contributed by Jeff Hankins, 20-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Theoremcnpfcf 18026* A function is continuous at point iff respects cluster points there. (Contributed by Jeff Hankins, 14-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
TopOn TopOn

Theoremcnfcf 18027* Continuity of a function in terms of cluster points of a function. (Contributed by Jeff Hankins, 28-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
TopOn TopOn

Theoremalexsublem 18028* Lemma for alexsub 18029. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremalexsub 18029* The Alexander Subbase Theorem: If is a subbase for the topology , and any cover taken from has a finite subcover, then the generated topology is compact. This proof uses the ultrafilter lemma; see alexsubALT 18035 for a proof using Zorn's lemma. (Contributed by Jeff Hankins, 24-Jan-2010.) (Revised by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremalexsubb 18030* Biconditional form of the Alexander Subbase Theorem alexsub 18029. (Contributed by Mario Carneiro, 27-Aug-2015.)
UFL

TheoremalexsubALTlem1 18031* Lemma for alexsubALT 18035. A compact space has a subbase such that every cover taken from it has a finite subcover. (Contributed by Jeff Hankins, 27-Jan-2010.)

TheoremalexsubALTlem2 18032* Lemma for alexsubALT 18035. Every subset of a base which has no finite subcover is a subset of a maximal such collection. (Contributed by Jeff Hankins, 27-Jan-2010.)

TheoremalexsubALTlem3 18033* Lemma for alexsubALT 18035. If a point is covered by a collection taken from the base with no finite subcover, a set from the subbase can be added that covers the point so that the resulting collection has no finite subcover. (Contributed by Jeff Hankins, 28-Jan-2010.) (Revised by Mario Carneiro, 14-Dec-2013.)

TheoremalexsubALTlem4 18034* Lemma for alexsubALT 18035. If any cover taken from a subbase has a finite subcover, any cover taken from the corresponding base has a finite subcover. (Contributed by Jeff Hankins, 28-Jan-2010.) (Revised by Mario Carneiro, 14-Dec-2013.)

TheoremalexsubALT 18035* The Alexander Subbase Theorem: a space is compact iff it has a subbase such that any cover taken from the subbase has a finite subcover. (Contributed by Jeff Hankins, 24-Jan-2010.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremptcmplem1 18036* Lemma for ptcmp 18042. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremptcmplem2 18037* Lemma for ptcmp 18042. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremptcmplem3 18038* Lemma for ptcmp 18042. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremptcmplem4 18039* Lemma for ptcmp 18042. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremptcmplem5 18040* Lemma for ptcmp 18042. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremptcmpg 18041 Tychonoff's theorem: The product of compact spaces is compact. The choice principles needed are encoded in the last hypothesis: the base set of the product must be well-orderable and satisfy the ultrafilter lemma. Both these assumptions are satisfied if is well-orderable, so if we assume the Axiom of Choice we can eliminate them (see ptcmp 18042). (Contributed by Mario Carneiro, 27-Aug-2015.)
UFL

Theoremptcmp 18042 Tychonoff's theorem: The product of compact spaces is compact. The proof uses the Axiom of Choice. (Contributed by Mario Carneiro, 26-Aug-2015.)

11.2.5  Extension by continuity

Syntaxccnext 18043 Extend class notation with the continuous extension operation.
CnExt

Definitiondf-cnext 18044* Define the continuous extension of a given function. (Contributed by Thierry Arnoux, 1-Dec-2017.)
CnExt t

Theoremcnextval 18045* The function applying continuous extension to a given function . (Contributed by Thierry Arnoux, 1-Dec-2017.)
CnExt t

Theoremcnextfval 18046* The continuous extension of a given function . (Contributed by Thierry Arnoux, 1-Dec-2017.)
CnExt t

Theoremcnextrel 18047 In the general case, a continuous extension is a relation. (Contributed by Thierry Arnoux, 20-Dec-2017.)
CnExt

Theoremcnextfun 18048 If the target space is Hausdorff, a continuous extension is a function (Contributed by Thierry Arnoux, 20-Dec-2017.)
CnExt

Theoremcnextfvval 18049* The value of the continuous extension of a given function at a point . (Contributed by Thierry Arnoux, 21-Dec-2017.)
t        CnExt t

Theoremcnextf 18050* Extension by continuity. The extension by continuity is a function. (Contributed by Thierry Arnoux, 25-Dec-2017.)
t        CnExt

Theoremcnextcn 18051* Extension by continuity. Theorem 1 of [BourbakiTop1] p. I.57. Given a topology on , a subset dense in , this states a condition for from to a regular space to be extensible by continuity (Contributed by Thierry Arnoux, 1-Jan-2018.)
t               CnExt

Theoremcnextfres 18052* and its extension by continuity agree on the domain of . (Contributed by Thierry Arnoux, 17-Jan-2018.)
t               t        CnExt

11.2.6  Topological groups

Syntaxctmd 18053 Extend class notation with the class of all topological monoids.
TopMnd

Syntaxctgp 18054 Extend class notation with the class of all topological groups.

Definitiondf-tmd 18055* Define the class of all topological monoids. A topological monoid is a monoid whose operation is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
TopMnd

Definitiondf-tgp 18056* Define the class of all topological groups. A topological group is a group whose operation and inverse function are continuous. (Contributed by FL, 18-Apr-2010.)
TopMnd

Theoremistmd 18057 The predicate "is a topological monoid". (Contributed by Mario Carneiro, 19-Sep-2015.)
TopMnd

Theoremtmdmnd 18058 A topological monoid is a monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
TopMnd

Theoremtmdtps 18059 A topological monoid is a topological space. (Contributed by Mario Carneiro, 19-Sep-2015.)
TopMnd

Theoremistgp 18060 The predicate "is a topological group". Definition of [BourbakiTop1] p. III.1 (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
TopMnd

Theoremtgpgrp 18061 A topological group is a group. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)

Theoremtgptmd 18062 A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
TopMnd

Theoremtgptps 18063 A topological group is a topological space. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)

Theoremtmdtopon 18064 The topology of a topological monoid. (Contributed by Mario Carneiro, 27-Jun-2014.) (Revised by Mario Carneiro, 13-Aug-2015.)
TopMnd TopOn

Theoremtgptopon 18065 The topology of a topological group. (Contributed by Mario Carneiro, 27-Jun-2014.) (Revised by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtmdcn 18066 In a topological monoid, the operation representing the functionalization of the operator slot is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
TopMnd

Theoremtgpcn 18067 In a topological group, the operation representing the functionalization of the operator slot is continuous. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)

Theoremtgpinv 18068 In a topological group, the inverse function is continuous. (Contributed by FL, 21-Jun-2010.) (Revised by FL, 27-Jun-2014.)

Theoremgrpinvhmeo 18069 The inverse function in a topological group is a homeomorphism from the group to itself. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremcnmpt1plusg 18070* Continuity of the group sum; analogue of cnmpt12f 17651 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
TopMnd       TopOn

Theoremcnmpt2plusg 18071* Continuity of the group sum; analogue of cnmpt22f 17660 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
TopMnd       TopOn       TopOn

Theoremtmdcn2 18072* Write out the definition of continuity of explicitly. (Contributed by Mario Carneiro, 20-Sep-2015.)
TopMnd

Theoremtgpsubcn 18073 In a topological group, the "subtraction" (or "division") is continuous. Axiom GT' of [BourbakiTop1] p. III.1 (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 19-Mar-2015.)

Theoremistgp2 18074 A group with a topology is a topological group iff the subtraction operation is continuous. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtmdmulg 18075* In a topological monoid, the n-times group multiple function is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
.g              TopMnd

Theoremtgpmulg 18076* In a topological group, the n-times group multiple function is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
.g

Theoremtgpmulg2 18077 In a topological monoid, the group multiple function is jointly continuous (although this is not saying much as one of the factors is discrete). Use zdis 18800 to write the left topology as a subset of the complexes. (Contributed by Mario Carneiro, 19-Sep-2015.)
.g

Theoremtmdgsum 18078* In a topological monoid, the group sum operation is a continuous function from the function space to the base topology. This theorem is not true when is infinite, because in this case for any basic open set of the domain one of the factors will be the whole space, so by varying the value of the functions to sum at this index, one can achieve any desired sum. (Contributed by Mario Carneiro, 19-Sep-2015.)
CMnd TopMnd g

Theoremtmdgsum2 18079* For any neighborhood of , there is a neighborhood of such that any sum of elements in sums to an element of . (Contributed by Mario Carneiro, 19-Sep-2015.)
.g       CMnd       TopMnd                                   g

Theoremoppgtmd 18080 The opposite of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
oppg       TopMnd TopMnd

Theoremoppgtgp 18081 The opposite of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
oppg

Theoremdistgp 18082 Any group equipped with the discrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremindistgp 18083 Any group equipped with the indiscrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremsymgtgp 18084 The symmetric group is a topological group. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtmdlactcn 18085* The left group action of element in a topological monoid is a continuous function. (Contributed by FL, 18-Mar-2008.) (Revised by Mario Carneiro, 14-Aug-2015.)
TopMnd

Theoremtgplacthmeo 18086* The left group action of element in a topological group is a homeomorphism from the group to itself. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremsubmtmd 18087 A submonoid of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 6-Oct-2015.)
s        TopMnd SubMnd TopMnd

Theoremsubgtgp 18088 A subgroup of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
s        SubGrp

Theoremsubgntr 18089 A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 18091, the interior of a subgroup is either a subgroup, or empty. (Contributed by Mario Carneiro, 19-Sep-2015.)
SubGrp

Theoremopnsubg 18090 An open subgroup of a topological group is also closed. (Contributed by Mario Carneiro, 17-Sep-2015.)
SubGrp

Theoremclssubg 18091 The closure of a subgroup in a topological group is a subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
SubGrp SubGrp

Theoremclsnsg 18092 The closure of a normal subgroup is a normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
NrmSGrp NrmSGrp

Theoremcldsubg 18093 A subgroup of finite index is closed iff it is open. (Contributed by Mario Carneiro, 20-Sep-2015.)
~QG               SubGrp

Theoremtgpconcompeqg 18094* The connected component containing is the left coset of the identity component containing . (Contributed by Mario Carneiro, 17-Sep-2015.)
t        ~QG        t

Theoremtgpconcomp 18095* The identity component, the connected component containing the identity element, is a closed (concompcld 17450) normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
t        NrmSGrp

Theoremtgpconcompss 18096* The identity component is a subset of any open subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
t        SubGrp

Theoremghmcnp 18097 A group homomorphism on topological groups is continuous everywhere if it is continuous at any point. (Contributed by Mario Carneiro, 21-Oct-2015.)
TopMnd TopMnd

Theoremsnclseqg 18098 The coset of the closure of the identity is the closure of a point. (Contributed by Mario Carneiro, 22-Sep-2015.)
~QG

Theoremtgphaus 18099 A topological group is Hausdorff iff the identity subgroup is closed. (Contributed by Mario Carneiro, 18-Sep-2015.)

Theoremtgpt1 18100 Hausdorff and T1 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.)

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