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

Theoremflimelbas 18001 A limit point of a filter belongs to its base set. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 9-Apr-2015.)

Theoremflimfil 18002 Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremflimtopon 18003 Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.)
TopOn

Theoremelflim 18004 The predicate "is a limit point of a filter." (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 23-Aug-2015.)
TopOn

Theoremflimss2 18005 A limit point of a filter is a limit point of a finer filter. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
TopOn

Theoremflimss1 18006 A limit point of a filter is a limit point in a coarser topology. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
TopOn

Theoremneiflim 18007 A point is a limit point of its neighborhood filter. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
TopOn

Theoremflimopn 18008* The condition for being a limit point of a filter still holds if one only considers open neighborhoods. (Contributed by Jeff Hankins, 4-Sep-2009.) (Proof shortened by Mario Carneiro, 9-Apr-2015.)
TopOn

Theoremfbflim 18009* A condition for a filter to converge to a point involving one of its bases. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
TopOn

Theoremfbflim2 18010* A condition for a filter base to converge to a point . Use neighborhoods instead of open neighborhoods. Compare fbflim 18009. (Contributed by FL, 4-Jul-2011.) (Revised by Stefan O'Rear, 6-Aug-2015.)
TopOn

Theoremflimclsi 18011 The convergent points of a filter are a subset of the closure of any of the filter sets. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Theoremhausflimlem 18012 If and are both limits of the same filter, then all neighborhoods of and intersect. (Contributed by Mario Carneiro, 21-Sep-2015.)

Theoremhausflimi 18013* One direction of hausflim 18014. A filter in a Hausdorf space has at most one limit. (Contributed by FL, 14-Nov-2010.) (Revised by Mario Carneiro, 21-Sep-2015.)

Theoremhausflim 18014* A condition for a topology to be Hausdorff in terms of filters. A topology is Hausdorff iff every filter has at most one limit point. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremflimcf 18015* Fineness is properly characterized by the property that every limit point of a filter in the finer topology is a limit point in the coarser topology. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 23-Aug-2015.)
TopOn TopOn

Theoremflimrest 18016 The set of limit points in a restricted topological space. (Contributed by Mario Carneiro, 15-Oct-2015.)
TopOn t t

Theoremflimclslem 18017 Lemma for flimcls 18018. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)
TopOn

Theoremflimcls 18018* Closure in terms of filter convergence. (Contributed by Jeff Hankins, 28-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
TopOn

Theoremflimsncls 18019 If is a limit point of the filter , then all the points which specialize (in the specialization preorder) are also limit points. Thus, the set of limit points is a union of closed sets (although this is only nontrivial for non-T1 spaces). (Contributed by Mario Carneiro, 20-Sep-2015.)

Theoremhauspwpwf1 18020* Lemma for hauspwpwdom 18021. Points in the closure of a set in a Hausdorff space are characterized by the open neighborhoods they extend into the generating set. (Contributed by Mario Carneiro, 28-Jul-2015.)

Theoremhauspwpwdom 18021 If is a Hausdorff space, then the cardinality of the closure of a set is bounded by the double powerset of . In particular, a Hausdorff space with a dense subset has cardinality at most , and a separable Hausdorff space has cardinality at most . (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 28-Jul-2015.)

Theoremflffval 18022* Given a topology and a filtered set, return the convergence function on the functions from the filtered set to the base set of the topological space. (Contributed by Jeff Hankins, 14-Oct-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (Revised by Stefan O'Rear, 6-Aug-2015.)
TopOn

Theoremflfval 18023 Given a function from a filtered set to a topological space, define the set of limit points of the function. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
TopOn

Theoremflfnei 18024* The property of being a limit point of a function in terms of neighborhoods. (Contributed by Jeff Hankins, 9-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
TopOn

Theoremflfneii 18025* A neighborhood of a limit point of a function contains the image of a filter element. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremisflf 18026* 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 18027 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 18028* 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 18029* Limit points of a function can be defined using topological bases. (Contributed by Mario Carneiro, 19-Sep-2015.)
TopOn

Theoremhausflf 18030* 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 18031 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 18032 Forward direction of cnpflf 18034. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Theoremcnpflf2 18033 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 18034* 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 18035* 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 18036* 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 18037 A continuous function preserves filter limits. (Contributed by Mario Carneiro, 18-Sep-2015.)
TopOn

Theoremlmflf 18038 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 18039* Two sequences converge in a filter iff the sequence of their ordered pairs converges. (Contributed by Mario Carneiro, 19-Sep-2015.)
TopOn       TopOn

Theoremflfcnp2 18040* 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 18041* 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 18042* A cluster point of a filter. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfclsfil 18043 Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfclstop 18044 Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfclstopon 18045 Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.)
TopOn

Theoremisfcls2 18046* A cluster point of a filter. (Contributed by Mario Carneiro, 26-Aug-2015.)
TopOn

Theoremfclsopn 18047* 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 18048 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 18049 A cluster point is in the base set. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)

Theoremfclsneii 18050 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 18051 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 18052* Cluster points in terms of neighborhoods. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
TopOn

Theoremsupnfcls 18053* 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 18054* Cluster points in terms of filter bases. (Contributed by Jeff Hankins, 13-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
TopOn

Theoremfclsss1 18055 A finer topology has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
TopOn

Theoremfclsss2 18056 A finer filter has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
TopOn

Theoremfclsrest 18057 The set of cluster points in a restricted topological space. (Contributed by Mario Carneiro, 15-Oct-2015.)
TopOn t t

Theoremfclscf 18058* 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 18059 A limit point is a cluster point. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfclsfnflim 18060* 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 18061* A filter converges to a point iff every finer filter clusters there. Along with fclsfnflim 18060, 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 18062 Forward direction of fclscmp 18063. Every filter clusters in a compact space. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfclscmp 18063* 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 18064 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 18065* 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 18066 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 18067* 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 18068* 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 18069 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 18070 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 18071 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 18072 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 18073 Lemma for cnpfcf 18074. 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 18074* 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 18075* 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 18076* Lemma for alexsub 18077. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremalexsub 18077* 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 18083 for a proof using Zorn's lemma. (Contributed by Jeff Hankins, 24-Jan-2010.) (Revised by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremalexsubb 18078* Biconditional form of the Alexander Subbase Theorem alexsub 18077. (Contributed by Mario Carneiro, 27-Aug-2015.)
UFL

TheoremalexsubALTlem1 18079* Lemma for alexsubALT 18083. 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 18080* Lemma for alexsubALT 18083. 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 18081* Lemma for alexsubALT 18083. 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 18082* Lemma for alexsubALT 18083. 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 18083* 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 18084* Lemma for ptcmp 18090. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremptcmplem2 18085* Lemma for ptcmp 18090. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremptcmplem3 18086* Lemma for ptcmp 18090. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremptcmplem4 18087* Lemma for ptcmp 18090. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremptcmplem5 18088* Lemma for ptcmp 18090. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremptcmpg 18089 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 18090). (Contributed by Mario Carneiro, 27-Aug-2015.)
UFL

Theoremptcmp 18090 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 18091 Extend class notation with the continuous extension operation.
CnExt

Definitiondf-cnext 18092* Define the continuous extension of a given function. (Contributed by Thierry Arnoux, 1-Dec-2017.)
CnExt t

Theoremcnextval 18093* The function applying continuous extension to a given function . (Contributed by Thierry Arnoux, 1-Dec-2017.)
CnExt t

Theoremcnextfval 18094* The continuous extension of a given function . (Contributed by Thierry Arnoux, 1-Dec-2017.)
CnExt t

Theoremcnextrel 18095 In the general case, a continuous extension is a relation. (Contributed by Thierry Arnoux, 20-Dec-2017.)
CnExt

Theoremcnextfun 18096 If the target space is Hausdorff, a continuous extension is a function (Contributed by Thierry Arnoux, 20-Dec-2017.)
CnExt

Theoremcnextfvval 18097* The value of the continuous extension of a given function at a point . (Contributed by Thierry Arnoux, 21-Dec-2017.)
t        CnExt t

Theoremcnextf 18098* Extension by continuity. The extension by continuity is a function. (Contributed by Thierry Arnoux, 25-Dec-2017.)
t        CnExt

Theoremcnextcn 18099* 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 18100* and its extension by continuity agree on the domain of . (Contributed by Thierry Arnoux, 17-Jan-2018.)
t               t        CnExt

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