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Theorem List for Metamath Proof Explorer - 17901-18000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisflf 17901* The property of being a limit point of a function. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
TopOn

Theoremflfelbas 17902 A limit point of a function is in the topological space. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
TopOn

Theoremflffbas 17903* Limit points of a function can be defined using filter bases. (Contributed by Jeff Hankins, 9-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
TopOn

Theoremflftg 17904* Limit points of a function can be defined using topological bases. (Contributed by Mario Carneiro, 19-Sep-2015.)
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Theoremhausflf 17905* If a function has its values in a Hausdorff space, then it has at most one limit value. (Contributed by FL, 14-Nov-2010.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremhausflf2 17906 If a convergent function has its values in a Hausdorff space, then it has a unique limit. (Contributed by FL, 14-Nov-2010.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremcnpflfi 17907 Forward direction of cnpflf 17909. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Theoremcnpflf2 17908 is continous at point iff a limit of when tends to is . Proposition 9 of [BourbakiTop1] p. TG I.50. (Contributed by FL, 29-May-2011.) (Revised by Mario Carneiro, 9-Apr-2015.)
TopOn TopOn

Theoremcnpflf 17909* Continuity of a function at a point in terms of filter limits. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
TopOn TopOn

Theoremcnflf 17910* A function is continuous iff it respects filter limits. (Contributed by Jeff Hankins, 6-Sep-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
TopOn TopOn

Theoremcnflf2 17911* A function is continuous iff it respects filter limits. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
TopOn TopOn

Theoremflfcnp 17912 A continuous function preserves filter limits. (Contributed by Mario Carneiro, 18-Sep-2015.)
TopOn

Theoremlmflf 17913 The topological limit relation on functions can be written in terms of the filter limit along the filter generated by the upper integer sets. (Contributed by Mario Carneiro, 13-Oct-2015.)
TopOn

Theoremtxflf 17914* Two sequences converge in a filter iff the sequence of their ordered pairs converges. (Contributed by Mario Carneiro, 19-Sep-2015.)
TopOn       TopOn

Theoremflfcnp2 17915* The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 19-Sep-2015.)
TopOn       TopOn

Theoremfclsval 17916* The set of all cluster points of a filter. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremisfcls 17917* A cluster point of a filter. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfclsfil 17918 Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfclstop 17919 Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfclstopon 17920 Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.)
TopOn

Theoremisfcls2 17921* A cluster point of a filter. (Contributed by Mario Carneiro, 26-Aug-2015.)
TopOn

Theoremfclsopn 17922* Write the cluster point condition in terms of open sets. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
TopOn

Theoremfclsopni 17923 An open neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfclselbas 17924 A cluster point is in the base set. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)

Theoremfclsneii 17925 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 17926 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 17927* Cluster points in terms of neighborhoods. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
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Theoremsupnfcls 17928* 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 17929* Cluster points in terms of filter bases. (Contributed by Jeff Hankins, 13-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
TopOn

Theoremfclsss1 17930 A finer topology has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
TopOn

Theoremfclsss2 17931 A finer filter has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
TopOn

Theoremfclsrest 17932 The set of cluster points in a restricted topological space. (Contributed by Mario Carneiro, 15-Oct-2015.)
TopOn t t

Theoremfclscf 17933* 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 17934 A limit point is a cluster point. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfclsfnflim 17935* 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 17936* A filter converges to a point iff every finer filter clusters there. Along with fclsfnflim 17935, 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 17937 Forward direction of fclscmp 17938. Every filter clusters in a compact space. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfclscmp 17938* A space is compact iff every filter clusters. (Contributed by Jeff Hankins, 20-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
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Theoremuffclsflim 17939 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 17940* 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 17941 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 17942* 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 17943* 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 17944 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 17945 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 17946 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 17947 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 17948 Lemma for cnpfcf 17949. 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 17949* 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 17950* 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 17951* Lemma for alexsub 17952. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremalexsub 17952* 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 17958 for a proof using Zorn's lemma. (Contributed by Jeff Hankins, 24-Jan-2010.) (Revised by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremalexsubb 17953* Biconditional form of the Alexander Subbase Theorem alexsub 17952. (Contributed by Mario Carneiro, 27-Aug-2015.)
UFL

TheoremalexsubALTlem1 17954* Lemma for alexsubALT 17958. 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 17955* Lemma for alexsubALT 17958. 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 17956* Lemma for alexsubALT 17958. 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 17957* Lemma for alexsubALT 17958. 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 17958* 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 17959* Lemma for ptcmp 17965. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremptcmplem2 17960* Lemma for ptcmp 17965. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremptcmplem3 17961* Lemma for ptcmp 17965. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremptcmplem4 17962* Lemma for ptcmp 17965. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremptcmplem5 17963* Lemma for ptcmp 17965. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremptcmpg 17964 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 17965). (Contributed by Mario Carneiro, 27-Aug-2015.)
UFL

Theoremptcmp 17965 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  Topological groups

Syntaxctmd 17966 Extend class notation with the class of all topological monoids.
TopMnd

Syntaxctgp 17967 Extend class notation with the class of all topological groups.

Definitiondf-tmd 17968* 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 17969* 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 17970 The predicate "is a topological monoid". (Contributed by Mario Carneiro, 19-Sep-2015.)
TopMnd

Theoremtmdmnd 17971 A topological monoid is a monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
TopMnd

Theoremtmdtps 17972 A topological monoid is a topological space. (Contributed by Mario Carneiro, 19-Sep-2015.)
TopMnd

Theoremistgp 17973 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 17974 A topological group is a group. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)

Theoremtgptmd 17975 A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
TopMnd

Theoremtgptps 17976 A topological group is a topological space. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)

Theoremtmdtopon 17977 The topology of a topological monoid. (Contributed by Mario Carneiro, 27-Jun-2014.) (Revised by Mario Carneiro, 13-Aug-2015.)
TopMnd TopOn

Theoremtgptopon 17978 The topology of a topological group. (Contributed by Mario Carneiro, 27-Jun-2014.) (Revised by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtmdcn 17979 In a topological monoid, the operation representing the functionalization of the operator slot is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
TopMnd

Theoremtgpcn 17980 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 17981 In a topological group, the inverse function is continuous. (Contributed by FL, 21-Jun-2010.) (Revised by FL, 27-Jun-2014.)

Theoremgrpinvhmeo 17982 The inverse function in a topological group is a homeomorphism from the group to itself. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremcnmpt1plusg 17983* Continuity of the group sum; analogue of cnmpt12f 17577 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
TopMnd       TopOn

Theoremcnmpt2plusg 17984* Continuity of the group sum; analogue of cnmpt22f 17586 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
TopMnd       TopOn       TopOn

Theoremtmdcn2 17985* Write out the definition of continuity of explicitly. (Contributed by Mario Carneiro, 20-Sep-2015.)
TopMnd

Theoremtgpsubcn 17986 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 17987 A group with a topology is a topological group iff the subtraction operation is continuous. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtmdmulg 17988* In a topological monoid, the n-times group multiple function is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
.g              TopMnd

Theoremtgpmulg 17989* In a topological group, the n-times group multiple function is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
.g

Theoremtgpmulg2 17990 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 18535 to write the left topology as a subset of the complexes. (Contributed by Mario Carneiro, 19-Sep-2015.)
.g

Theoremtmdgsum 17991* 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 17992* 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 17993 The opposite of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
oppg       TopMnd TopMnd

Theoremoppgtgp 17994 The opposite of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
oppg

Theoremdistgp 17995 Any group equipped with the discrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremindistgp 17996 Any group equipped with the indiscrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremsymgtgp 17997 The symmetric group is a topological group. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtmdlactcn 17998* 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 17999* 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 18000 A submonoid of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 6-Oct-2015.)
s        TopMnd SubMnd TopMnd

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