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Theorem List for Metamath Proof Explorer - 17201-17300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrestlp 17201 The limit points of a subset restrict naturally in a subspace. (Contributed by Mario Carneiro, 25-Dec-2016.)
t

Theoremrestperf 17202 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 17203 An open subset of a perfect space is perfect. (Contributed by Mario Carneiro, 25-Dec-2016.)
t        Perf Perf

Theoremresstopn 17204 The topology of a restricted structure. (Contributed by Mario Carneiro, 26-Aug-2015.)
s               t

Theoremresstps 17205 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 17206* Lemma for ordtbas 17210. In a total order, unbounded-above intervals are closed under intersection. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremordtval 17207* Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtuni 17208* Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremordtbas2 17209* Lemma for ordtbas 17210. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremordtbas 17210* 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 17211 Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop TopOn

Theoremordtopn1 17212* An upward ray is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtopn2 17213* A downward ray is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtopn3 17214* An open interval is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtcld1 17215* A downward ray is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtcld2 17216* An upward ray is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtcld3 17217* A closed interval is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordttop 17218 The order topology is a topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtcnv 17219 The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop ordTop

Theoremordtrest 17220 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 17221* Lemma for ordtrest2 17222. (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop

Theoremordtrest2 17222* 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 17223 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop TopOn

Theoremletop 17224 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremletopuni 17225 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremxrstopn 17226 The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
ordTop

Theoremxrstps 17227 The extended real number structure is a topological space. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremleordtvallem1 17228* Lemma for leordtval 17231. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremleordtvallem2 17229* Lemma for leordtval 17231. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremleordtval2 17230 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremleordtval 17231 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremiccordt 17232 A closed interval is closed in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremiocpnfordt 17233 An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremicomnfordt 17234 An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremiooordt 17235 An open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremreordt 17236 The real numbers are an open set in the topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremlecldbas 17237 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 17238* A neighborhood of contains an unbounded interval based at a real number. Together with xrtgioo 18790 (which describes neighborhoods of ) and mnfnei 17239, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 17235 and similar ensure that it has all the sets we want). (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremmnfnei 17239* A neighborhood of contains an unbounded interval based at a real number. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtrestixx 17240* 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 17241 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 17242 Extend class notation with the set of continuous functions between topologies.

Syntaxccnp 17243 Extend class notation with the set of functions between topologies continuous at a point.

Syntaxclm 17244 Extend class notation with a function on topological spaces whose value is the convergence relation for limit sequences in the space.

Definitiondf-cn 17245* 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 17253 for the predicate form. (Contributed by NM, 17-Oct-2006.)

Definitiondf-cnp 17246* 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 17247* 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 5765, and we use the otherwise unnecessary conjunct to ensure that. (Contributed by NM, 7-Sep-2006.)

Theoremlmrel 17248 The topological space convergence relation is a relation. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)

Theoremlmrcl 17249 Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.)

Theoremlmfval 17250* 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 17251* 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 17252* 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 17253* 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 17254* 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 17255* 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 17256* 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 17257* 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 17258 Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcntop2 17259 Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcnptop1 17260 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcnptop2 17261 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremiscnp3 17262* The predicate " is a continuous function from topology to topology at point ." (Contributed by NM, 15-May-2007.)
TopOn TopOn

Theoremcnprcl 17263 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcnf 17264 A continuous function is a mapping. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Theoremcnpf 17265 A continuous function at point is a mapping. (Contributed by FL, 17-Nov-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Theoremcnpcl 17266 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 17267 A continuous function is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremcnpf2 17268 A continuous function at point is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremcnprcl2 17269 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
TopOn

Theoremtgcn 17270* 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 17271* 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 17272* 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 17273 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 17274* Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.)

Theoremidcn 17275 A restricted identity function is a continuous function. (Contributed by FL, 27-Dec-2006.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
TopOn

Theoremlmbr 17276* 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 17247. (Contributed by Mario Carneiro, 14-Nov-2013.)
TopOn

Theoremlmbr2 17277* 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 17278* Express the binary relation "sequence converges to point " in a metric space using an arbitrary set of upper integers. This version of lmbr2 17277 presupposes that is a function. (Contributed by Mario Carneiro, 14-Nov-2013.)
TopOn

Theoremlmconst 17279 A constant sequence converges to its value. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)
TopOn

Theoremlmcvg 17280* Convergence property of a converging sequence. (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremiscnp4 17281* The predicate " is a continuous function from topology to topology at point ." in terms of neighborhoods. (Contributed by FL, 18-Jul-2011.) (Revised by Mario Carneiro, 10-Sep-2015.)
TopOn TopOn

Theoremcnpnei 17282* A condition for continuity at a point in terms of neighborhoods. (Contributed by Jeff Hankins, 7-Sep-2009.)

Theoremcnima 17283 An open subset of the codomain of a continuous function has an open preimage. (Contributed by FL, 15-Dec-2006.)

Theoremcnco 17284 The composition of two continuous functions is a continuous function. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Theoremcnpco 17285 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 17286 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 17287* 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 17288 Property of the preimage of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremcnntri 17289 Property of the preimage of an interior. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremcnclsi 17290 Property of the image of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremcncls2 17291* Continuity in terms of closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn TopOn

Theoremcncls 17292* Continuity in terms of closure. (Contributed by Jeff Hankins, 1-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
TopOn TopOn

Theoremcnntr 17293* Continuity in terms of interior. (Contributed by Jeff Hankins, 2-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
TopOn TopOn

Theoremcnss1 17294 If the topology is finer than , then there are more continuous functions from than from . (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn

Theoremcnss2 17295 If the topology is finer than , then there are fewer continuous functions into than into from some other space. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn

Theoremcncnpi 17296 A continuous function is continuous at all points. One direction of Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)

Theoremcnsscnp 17297 The set of continuous functions is a subset of the set of continuous functions at a point. (Contributed by Raph Levien, 21-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Theoremcncnp 17298* A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 15-May-2007.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremcncnp2 17299* A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)

Theoremcnnei 17300* Continuity in terms of neighborhoods. (Contributed by Thierry Arnoux, 3-Jan-2018.)

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