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Theorem List for Metamath Proof Explorer - 16901-17000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrestsn 16901 The only subspace topology induced by the topology . (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
t

Theoremrestsn2 16902 The subspace topology induced by a singleton. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
TopOn t

Theoremrestcld 16903* A closed set of a subspace topology is a closed set of the original topology intersected with the subset. (Contributed by FL, 11-Jul-2009.) (Proof shortened by Mario Carneiro, 15-Dec-2013.)
t

Theoremrestcldi 16904 A closed set is closed in the subspace topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
t

Theoremrestcldr 16905 A set which is closed in the subspace topology induced by a closed set is closed in the original topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
t

Theoremrestopnb 16906 If is an open subset of the subspace base set , then any subset of is open iff it is open in . (Contributed by Mario Carneiro, 2-Mar-2015.)
t

Theoremssrest 16907 If is a finer topology than , then the subspace topologies induced by maintain this relationship. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
t t

Theoremrestopn2 16908 The if is open, then is open in iff it is an open subset of . (Contributed by Mario Carneiro, 2-Mar-2015.)
t

Theoremrestdis 16909 A subspace of a discrete topology is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.)
t

Theoremrestfpw 16910 The restriction of the set of finite subsets of is the set of finite subsets of . (Contributed by Mario Carneiro, 18-Sep-2015.)
t

Theoremrestcls 16911 A closure in a subspace topology. (Contributed by Jeff Hankins, 22-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)
t

Theoremrestntr 16912 An interior in a subspace topology. Willard in General Topology says that there is no analog of restcls 16911 for interiors. In some sense, that is true. (Contributed by Jeff Hankins, 23-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)
t

Theoremrestlp 16913 The limit points of a subset restrict naturally in a subspace. (Contributed by Mario Carneiro, 25-Dec-2016.)
t

Theoremrestperf 16914 Perfection of a subspace. Note that the term "perfect set" is reserved for closed sets which are perfect in the subspace topology. (Contributed by Mario Carneiro, 25-Dec-2016.)
t        Perf

Theoremperfopn 16915 An open subset of a perfect space is perfect. (Contributed by Mario Carneiro, 25-Dec-2016.)
t        Perf Perf

Theoremresstopn 16916 The topology of a restricted structure. (Contributed by Mario Carneiro, 26-Aug-2015.)
s               t

Theoremresstps 16917 A restricted topological space is a topological space. Note that this theorem would not be true if was defined directly in terms of the TopSet slot instead of the derived function. (Contributed by Mario Carneiro, 13-Aug-2015.)
s

11.1.8  Order topology

Theoremordtbaslem 16918* Lemma for ordtbas 16922. In a total order, unbounded-above intervals are closed under intersection. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremordtval 16919* Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtuni 16920* Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremordtbas2 16921* Lemma for ordtbas 16922. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremordtbas 16922* In a total order, the finite intersections of the open rays generates the set of open intervals, but no more - these four collections form a subbasis for the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremordttopon 16923 Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop TopOn

Theoremordtopn1 16924* An upward ray is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtopn2 16925* A downward ray is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtopn3 16926* An open interval is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtcld1 16927* A downward ray is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtcld2 16928* An upward ray is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtcld3 16929* An closed interval is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordttop 16930 The order topology is a topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtcnv 16931 The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop ordTop

Theoremordtrest 16932 The subspace topology of an order topology is in general finer than the topology generated by the restricted order, but we do have inclusion in one direction. (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop ordTopt

Theoremordtrest2lem 16933* Lemma for ordtrest2 16934. (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop

Theoremordtrest2 16934* An interval-closed set in a total order has the same subspace topology as the restricted order topology. (An interval-closed set is the same thing as an open or half-open or closed interval in , but in other sets like there are interval-closed sets like that are not intervals.) (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop ordTopt

Theoremletopon 16935 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop TopOn

Theoremletop 16936 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremletopuni 16937 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremxrstopn 16938 The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
ordTop

Theoremxrstps 16939 The extended real number structure is a topological space. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremleordtvallem1 16940* Lemma for leordtval 16943. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremleordtvallem2 16941* Lemma for leordtval 16943. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremleordtval2 16942 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremleordtval 16943 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremiccordt 16944 A closed interval is closed in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremiocpnfordt 16945 An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremicomnfordt 16946 An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremiooordt 16947 An open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremreordt 16948 The real numbers are an open set in the topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremlecldbas 16949 The set of closed intervals forms a closed subbasis for the topology on the extended reals. Since our definition of a basis is in terms of open sets, we express this by showing that the complements of closed intervals form an open subbasis for the topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theorempnfnei 16950* A neighborhood of contains an unbounded interval based at a real number. Together with xrtgioo 18312 (which describes neighborhoods of ) and mnfnei 16951, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 16947 and similar ensure that it has all the sets we want). (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremmnfnei 16951* A neighborhood of contains an unbounded interval based at a real number. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtrestixx 16952* The restriction of the less than order to an interval gives the same topology as the subspace topology. (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop t ordTop

Theoremordtresticc 16953 The restriction of the less than order to a closed interval gives the same topology as the subspace topology. (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop t ordTop

11.1.9  Limits and continuity in topological spaces

Syntaxccn 16954 Extend class notation with the set of continuous functions between topologies.

Syntaxccnp 16955 Extend class notation with the set of functions between topologies continuous at a point.

Syntaxclm 16956 Extend class notation with a function on topological spaces whose value is the convergence relation for limit sequences in the space.

Definitiondf-cn 16957* Define a function on two topologies whose value is the set of continuous mappings from the first topology to the second. Based on definition of continuous function in [Munkres] p. 102. See iscn 16965 for the predicate form. (Contributed by NM, 17-Oct-2006.)

Definitiondf-cnp 16958* Define a function on two topologies whose value is the set of continuous mappings at a specified point in the first topology. Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 17-Oct-2006.)

Definitiondf-lm 16959* Define a function on topologies whose value is the convergence relation for the space. Although is typically a function from upper integers to the topological space, it doesn't have to be. Unfortunately, the value of the function must exist to use fvmpt 5602, and we use the otherwise unnecessary conjunct to ensure that. (Contributed by NM, 7-Sep-2006.)

Theoremlmrel 16960 The topological space convergence relation is a relation. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)

Theoremlmrcl 16961 Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.)

Theoremlmfval 16962* The relation "sequence converges to point " in a metric space. (Contributed by NM, 7-Sep-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn

Theoremcnfval 16963* The set of all continuous functions from topology to topology . (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremcnpfval 16964* The function mapping the points in a topology to the set of all functions from to topology continuous at that point. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremiscn 16965* The predicate " is a continuous function from topology to topology ." Definition of continuous function in [Munkres] p. 102. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremcnpval 16966* The set of all functions from topology to topology that are continuous at a point . (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
TopOn TopOn

Theoremiscnp 16967* The predicate " is a continuous function from topology to topology at point ." Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremiscn2 16968* The predicate " is a continuous function from topology to topology ." Definition of continuous function in [Munkres] p. 102. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremiscnp2 16969* The predicate " is a continuous function from topology to topology at point ." Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcntop1 16970 Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcntop2 16971 Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcnptop1 16972 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcnptop2 16973 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremiscnp3 16974* The predicate " is a continuous function from topology to topology at point ." (Contributed by NM, 15-May-2007.)
TopOn TopOn

Theoremcnprcl 16975 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcnf 16976 A continuous function is a mapping. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Theoremcnpf 16977 A continuous function at point is a mapping. (Contributed by FL, 17-Nov-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Theoremcnpcl 16978 The value of a continuous function from to at point belongs to the underlying set of topology . (Contributed by FL, 27-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Theoremcnf2 16979 A continuous function is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremcnpf2 16980 A continuous function at point is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremcnprcl2 16981 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
TopOn

Theoremtgcn 16982* The contininuity predicate when the range is given by a basis for a topology. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn              TopOn

Theoremtgcnp 16983* The "continuous at a point" predicate when the range is given by a basis for a topology. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn              TopOn

Theoremsubbascn 16984* The contininuity predicate when the range is given by a subbasis for a topology. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn                     TopOn

Theoremssidcn 16985 The identity function is a continuous function from one topology to another topology on the same set iff the domain is finer than the codomain. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremcnpimaex 16986* Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.)

Theoremidcn 16987 A restricted identity function is a continuous function. (Contributed by FL, 27-Dec-2006.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
TopOn

Theoremlmbr 16988* Express the binary relation "sequence converges to point " in a topological space. Definition 1.4-1 of [Kreyszig] p. 25. The condition allows us to use objects more general than sequences when convenient; see the comment in df-lm 16959. (Contributed by Mario Carneiro, 14-Nov-2013.)
TopOn

Theoremlmbr2 16989* Express the binary relation "sequence converges to point " in a metric space using an arbitrary set of upper integers. (Contributed by Mario Carneiro, 14-Nov-2013.)
TopOn

Theoremlmbrf 16990* Express the binary relation "sequence converges to point " in a metric space using an arbitrary set of upper integers. This version of lmbr2 16989 presupposes that is a function. (Contributed by Mario Carneiro, 14-Nov-2013.)
TopOn

Theoremlmconst 16991 A constant sequence converges to its value. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)
TopOn

Theoremlmcvg 16992* Convergence property of a converging sequence. (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremcnpnei 16993* A condition for continuity at a point in terms of neighborhoods. (Contributed by Jeff Hankins, 7-Sep-2009.)

Theoremcnima 16994 An open subset of the codomain of a continuous function has an open preimage. (Contributed by FL, 15-Dec-2006.)

Theoremcnco 16995 The composition of two continuous functions is a continuous function. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Theoremcnpco 16996 The composition of two continuous functions at point is a continuous function at point . Proposition of [BourbakiTop1] p. I.9. (Contributed by FL, 16-Nov-2006.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Theoremcnclima 16997 A closed subset of the codomain of a continuous function has a closed preimage. (Contributed by NM, 15-Mar-2007.) (Revised by Mario Carneiro, 21-Aug-2015.)

Theoremiscncl 16998* A definition of a continuous function using closed sets. Theorem 1 (d) of [BourbakiTop1] p. I.9. (Contributed by FL, 19-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremcncls2i 16999 Property of the preimage of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremcnntri 17000 Property of the preimage of an interior. (Contributed by Mario Carneiro, 25-Aug-2015.)

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