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

Theoremisfcf 18001* 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 18002* 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 18003 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 18004 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 18005 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 18006 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 18007 Lemma for cnpfcf 18008. 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 18008* 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 18009* 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 18010* Lemma for alexsub 18011. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremalexsub 18011* 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 18017 for a proof using Zorn's lemma. (Contributed by Jeff Hankins, 24-Jan-2010.) (Revised by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremalexsubb 18012* Biconditional form of the Alexander Subbase Theorem alexsub 18011. (Contributed by Mario Carneiro, 27-Aug-2015.)
UFL

TheoremalexsubALTlem1 18013* Lemma for alexsubALT 18017. 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 18014* Lemma for alexsubALT 18017. 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 18015* Lemma for alexsubALT 18017. 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 18016* Lemma for alexsubALT 18017. 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 18017* 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 18018* Lemma for ptcmp 18024. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremptcmplem2 18019* Lemma for ptcmp 18024. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremptcmplem3 18020* Lemma for ptcmp 18024. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremptcmplem4 18021* Lemma for ptcmp 18024. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremptcmplem5 18022* Lemma for ptcmp 18024. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremptcmpg 18023 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 18024). (Contributed by Mario Carneiro, 27-Aug-2015.)
UFL

Theoremptcmp 18024 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 18025 Extend class notation with the continuous extension operation.
CnExt

Definitiondf-cnext 18026* Define the continuous extension of a given function. (Contributed by Thierry Arnoux, 1-Dec-2017.)
CnExt t

Theoremcnextval 18027* The function applying continuous extension to a given function . (Contributed by Thierry Arnoux, 1-Dec-2017.)
CnExt t

Theoremcnextfval 18028* The continuous extension of a given function . (Contributed by Thierry Arnoux, 1-Dec-2017.)
CnExt t

Theoremcnextrel 18029 In the general case, a continuous extension is a relation. (Contributed by Thierry Arnoux, 20-Dec-2017.)
CnExt

Theoremcnextfun 18030 If the target space is Hausdorff, a continuous extension is a function (Contributed by Thierry Arnoux, 20-Dec-2017.)
CnExt

Theoremcnextfvval 18031* The value of the continuous extension of a given function at a point . (Contributed by Thierry Arnoux, 21-Dec-2017.)
t        CnExt t

Theoremcnextf 18032* Extension by continuity. The extension by continuity is a function. (Contributed by Thierry Arnoux, 25-Dec-2017.)
t        CnExt

Theoremcnextcn 18033* 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 18034* 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 18035 Extend class notation with the class of all topological monoids.
TopMnd

Syntaxctgp 18036 Extend class notation with the class of all topological groups.

Definitiondf-tmd 18037* 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 18038* 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 18039 The predicate "is a topological monoid". (Contributed by Mario Carneiro, 19-Sep-2015.)
TopMnd

Theoremtmdmnd 18040 A topological monoid is a monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
TopMnd

Theoremtmdtps 18041 A topological monoid is a topological space. (Contributed by Mario Carneiro, 19-Sep-2015.)
TopMnd

Theoremistgp 18042 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 18043 A topological group is a group. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)

Theoremtgptmd 18044 A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
TopMnd

Theoremtgptps 18045 A topological group is a topological space. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)

Theoremtmdtopon 18046 The topology of a topological monoid. (Contributed by Mario Carneiro, 27-Jun-2014.) (Revised by Mario Carneiro, 13-Aug-2015.)
TopMnd TopOn

Theoremtgptopon 18047 The topology of a topological group. (Contributed by Mario Carneiro, 27-Jun-2014.) (Revised by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtmdcn 18048 In a topological monoid, the operation representing the functionalization of the operator slot is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
TopMnd

Theoremtgpcn 18049 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 18050 In a topological group, the inverse function is continuous. (Contributed by FL, 21-Jun-2010.) (Revised by FL, 27-Jun-2014.)

Theoremgrpinvhmeo 18051 The inverse function in a topological group is a homeomorphism from the group to itself. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremcnmpt1plusg 18052* Continuity of the group sum; analogue of cnmpt12f 17633 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
TopMnd       TopOn

Theoremcnmpt2plusg 18053* Continuity of the group sum; analogue of cnmpt22f 17642 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
TopMnd       TopOn       TopOn

Theoremtmdcn2 18054* Write out the definition of continuity of explicitly. (Contributed by Mario Carneiro, 20-Sep-2015.)
TopMnd

Theoremtgpsubcn 18055 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 18056 A group with a topology is a topological group iff the subtraction operation is continuous. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtmdmulg 18057* In a topological monoid, the n-times group multiple function is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
.g              TopMnd

Theoremtgpmulg 18058* In a topological group, the n-times group multiple function is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
.g

Theoremtgpmulg2 18059 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 18732 to write the left topology as a subset of the complexes. (Contributed by Mario Carneiro, 19-Sep-2015.)
.g

Theoremtmdgsum 18060* 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 18061* 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 18062 The opposite of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
oppg       TopMnd TopMnd

Theoremoppgtgp 18063 The opposite of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
oppg

Theoremdistgp 18064 Any group equipped with the discrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremindistgp 18065 Any group equipped with the indiscrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremsymgtgp 18066 The symmetric group is a topological group. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtmdlactcn 18067* 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 18068* 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 18069 A submonoid of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 6-Oct-2015.)
s        TopMnd SubMnd TopMnd

Theoremsubgtgp 18070 A subgroup of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
s        SubGrp

Theoremsubgntr 18071 A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 18073, the interior of a subgroup is either a subgroup, or empty. (Contributed by Mario Carneiro, 19-Sep-2015.)
SubGrp

Theoremopnsubg 18072 An open subgroup of a topological group is also closed. (Contributed by Mario Carneiro, 17-Sep-2015.)
SubGrp

Theoremclssubg 18073 The closure of a subgroup in a topological group is a subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
SubGrp SubGrp

Theoremclsnsg 18074 The closure of a normal subgroup is a normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
NrmSGrp NrmSGrp

Theoremcldsubg 18075 A subgroup of finite index is closed iff it is open. (Contributed by Mario Carneiro, 20-Sep-2015.)
~QG               SubGrp

Theoremtgpconcompeqg 18076* The connected component containing is the left coset of the identity component containing . (Contributed by Mario Carneiro, 17-Sep-2015.)
t        ~QG        t

Theoremtgpconcomp 18077* The identity component, the connected component containing the identity element, is a closed (concompcld 17432) normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
t        NrmSGrp

Theoremtgpconcompss 18078* The identity component is a subset of any open subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
t        SubGrp

Theoremghmcnp 18079 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 18080 The coset of the closure of the identity is the closure of a point. (Contributed by Mario Carneiro, 22-Sep-2015.)
~QG

Theoremtgphaus 18081 A topological group is Hausdorff iff the identity subgroup is closed. (Contributed by Mario Carneiro, 18-Sep-2015.)

Theoremtgpt1 18082 Hausdorff and T1 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.)

Theoremtgpt0 18083 Hausdorff and T0 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.)

Theoremdivstgpopn 18084* A quotient map in a topological group is an open map. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG                             ~QG        NrmSGrp

Theoremdivstgplem 18085* Lemma for divstgp 18086. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG                             ~QG        ~QG        NrmSGrp

Theoremdivstgp 18086 The quotient of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
s ~QG        NrmSGrp

Theoremdivstgphaus 18087 The quotient of a topological group by a closed normal subgroup is a Hausdorff topological group. In particular, the quotient by the closure of the identity is a Hausdorff group, isomorphic to both the Kolmogorov quotient and the Hausdorff quotient operations on topological spaces (because T0 and Hausdorff coincide for topological groups). (Contributed by Mario Carneiro, 22-Sep-2015.)
s ~QG                      NrmSGrp

Theoremprdstmdd 18088 The product of a family of topological monoids is a topological monoid. (Contributed by Mario Carneiro, 22-Sep-2015.)
s                     TopMnd       TopMnd

Theoremprdstgpd 18089 The product of a family of topological groups is a topological group. (Contributed by Mario Carneiro, 22-Sep-2015.)
s

11.2.7  Infinite group sum on topological groups

Syntaxctsu 18090 Extend class notation to include infinite group sums in a topological group.
tsums

Definitiondf-tsms 18091* Define the set of limit points of an infinite group sum for the topological group . If is Hausdorff, then there will be at most one element in this set and tsums selects this unique element if it exists. tsums is a way to say that the sum exists and is unique. Note that unlike (df-sum 12421) and g (df-gsum 13669), this does not return the sum itself, but rather the set of all such sums, which is usually either empty or a singleton. (Contributed by Mario Carneiro, 2-Sep-2015.)
tsums g

Theoremtsmsfbas 18092* The collection of all sets of the form , which can be read as the set of all finite subsets of which contain as a subset, for each finite subset of , form a filter base. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtsmslem1 18093 The finite partial sums of a function are defined in a commutative monoid. (Contributed by Mario Carneiro, 2-Sep-2015.)
CMnd                     g

Theoremtsmsval2 18094* Definition of the topological group sum(s) of a collection of values in the group with index set . (Contributed by Mario Carneiro, 2-Sep-2015.)
tsums g

Theoremtsmsval 18095* Definition of the topological group sum(s) of a collection of values in the group with index set . (Contributed by Mario Carneiro, 2-Sep-2015.)
tsums g

Theoremtsmspropd 18096 The group sum depends only on the base set, additive operation, and topology components. Note that for entirely unrestricted functions, there can be dependency on out-of-domain values of the operation, so this is somewhat weaker than mndpropd 14662 etc. (Contributed by Mario Carneiro, 18-Sep-2015.)
tsums tsums

Theoremeltsms 18097* The property of being a sum of the sequence in the topological commutative monoid . (Contributed by Mario Carneiro, 2-Sep-2015.)
CMnd                            tsums g

Theoremtsmsi 18098* The property of being a sum of the sequence in the topological commutative monoid . (Contributed by Mario Carneiro, 2-Sep-2015.)
CMnd                            tsums                      g

Theoremtsmscl 18099 A sum in a topological group is an element of the group. (Contributed by Mario Carneiro, 2-Sep-2015.)
CMnd                            tsums

Theoremhaustsms 18100* A Hausdorff group has at most one limit point for a given sum. (Contributed by Mario Carneiro, 2-Sep-2015.)
CMnd                                          tsums

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