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

Theoremmaxlp 17201 A point is a limit point of the whole space iff the singleton of the point is not open. (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremclslp 17202 The closure of a subset of a topological space is the subset together with its limit points. Theorem 6.6 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.)

Theoremislpi 17203 A point belonging to a set's closure but not the set itself is a limit point. (Contributed by NM, 8-Nov-2007.)

Theoremcldlp 17204 A subset of a topological space is closed iff it contains all its limit points. Corollary 6.7 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.)

Theoremisperf 17205 Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Perf

Theoremisperf2 17206 Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Perf

Theoremisperf3 17207* A perfect space is a topology which has no open singletons. (Contributed by Mario Carneiro, 24-Dec-2016.)
Perf

Theoremperflp 17208 The limit points of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Perf

Theoremperfi 17209 Property of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Perf

Theoremperftop 17210 A perfect space is a topology. (Contributed by Mario Carneiro, 25-Dec-2016.)
Perf

11.1.7  Subspace topologies

Theoremrestrcl 17211 Reverse closure for the subspace topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
t

Theoremrestbas 17212 A subspace topology basis is a basis. is normally a subset of the base set of . (Contributed by Mario Carneiro, 19-Mar-2015.)
t

Theoremtgrest 17213 A subspace can be generated by restricted sets from a basis for the original topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Proof shortened by Mario Carneiro, 30-Aug-2015.)
t t

Theoremresttop 17214 A subspace topology is a topology. Definition of subspace topology in [Munkres] p. 89. is normally a subset of the base set of . (Contributed by FL, 15-Apr-2007.) (Revised by Mario Carneiro, 1-May-2015.)
t

Theoremresttopon 17215 A subspace topology is a topology on the base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn t TopOn

Theoremrestuni 17216 The underlying set of a subspace topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
t

Theoremstoig 17217 The topological space built with a subspace topology. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.)
TopSet t

Theoremrestco 17218 Composition of subspaces. (Contributed by Mario Carneiro, 15-Dec-2013.) (Revised by Mario Carneiro, 1-May-2015.)
t t t

Theoremrestabs 17219 Equivalence of being a subspace of a subspace and being a subspace of the original. (Contributed by Jeff Hankins, 11-Jul-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.)
t t t

Theoremrestin 17220 When the subspace region is not a subset of the base of the topology, the resulting set is the same as the subspace restricted to the base. (Contributed by Mario Carneiro, 15-Dec-2013.)
t t

Theoremrestuni2 17221 The underlying set of a subspace topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
t

Theoremresttopon2 17222 The underlying set of a subspace topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn t TopOn

Theoremrest0 17223 The subspace topology induced by the topology on the empty set. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 1-May-2015.)
t

Theoremrestsn 17224 The only subspace topology induced by the topology . (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
t

Theoremrestsn2 17225 The subspace topology induced by a singleton. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
TopOn t

Theoremrestcld 17226* 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 17227 A closed set is closed in the subspace topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
t

Theoremrestcldr 17228 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 17229 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 17230 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 17231 The if is open, then is open in iff it is an open subset of . (Contributed by Mario Carneiro, 2-Mar-2015.)
t

Theoremrestdis 17232 A subspace of a discrete topology is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.)
t

Theoremrestfpw 17233 The restriction of the set of finite subsets of is the set of finite subsets of . (Contributed by Mario Carneiro, 18-Sep-2015.)
t

Theoremneitr 17234 The neighborhood of a trace is the trace of the neighborhood. (Contributed by Thierry Arnoux, 17-Jan-2018.)
t t

Theoremrestcls 17235 A closure in a subspace topology. (Contributed by Jeff Hankins, 22-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)
t

Theoremrestntr 17236 An interior in a subspace topology. Willard in General Topology says that there is no analog of restcls 17235 for interiors. In some sense, that is true. (Contributed by Jeff Hankins, 23-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)
t

Theoremrestlp 17237 The limit points of a subset restrict naturally in a subspace. (Contributed by Mario Carneiro, 25-Dec-2016.)
t

Theoremrestperf 17238 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 17239 An open subset of a perfect space is perfect. (Contributed by Mario Carneiro, 25-Dec-2016.)
t        Perf Perf

Theoremresstopn 17240 The topology of a restricted structure. (Contributed by Mario Carneiro, 26-Aug-2015.)
s               t

Theoremresstps 17241 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 17242* Lemma for ordtbas 17246. In a total order, unbounded-above intervals are closed under intersection. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremordtval 17243* Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtuni 17244* Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremordtbas2 17245* Lemma for ordtbas 17246. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremordtbas 17246* 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 17247 Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop TopOn

Theoremordtopn1 17248* An upward ray is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtopn2 17249* A downward ray is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtopn3 17250* An open interval is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtcld1 17251* A downward ray is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtcld2 17252* An upward ray is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtcld3 17253* A closed interval is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordttop 17254 The order topology is a topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtcnv 17255 The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop ordTop

Theoremordtrest 17256 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 17257* Lemma for ordtrest2 17258. (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop

Theoremordtrest2 17258* 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 17259 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop TopOn

Theoremletop 17260 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremletopuni 17261 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremxrstopn 17262 The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
ordTop

Theoremxrstps 17263 The extended real number structure is a topological space. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremleordtvallem1 17264* Lemma for leordtval 17267. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremleordtvallem2 17265* Lemma for leordtval 17267. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremleordtval2 17266 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremleordtval 17267 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremiccordt 17268 A closed interval is closed in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremiocpnfordt 17269 An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremicomnfordt 17270 An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremiooordt 17271 An open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremreordt 17272 The real numbers are an open set in the topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremlecldbas 17273 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 17274* A neighborhood of contains an unbounded interval based at a real number. Together with xrtgioo 18827 (which describes neighborhoods of ) and mnfnei 17275, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 17271 and similar ensure that it has all the sets we want). (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremmnfnei 17275* A neighborhood of contains an unbounded interval based at a real number. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtrestixx 17276* 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 17277 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 17278 Extend class notation with the set of continuous functions between topologies.

Syntaxccnp 17279 Extend class notation with the set of functions between topologies continuous at a point.

Syntaxclm 17280 Extend class notation with a function on topological spaces whose value is the convergence relation for limit sequences in the space.

Definitiondf-cn 17281* 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 17289 for the predicate form. (Contributed by NM, 17-Oct-2006.)

Definitiondf-cnp 17282* 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 17283* 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 5798, and we use the otherwise unnecessary conjunct to ensure that. (Contributed by NM, 7-Sep-2006.)

Theoremlmrel 17284 The topological space convergence relation is a relation. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)

Theoremlmrcl 17285 Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.)

Theoremlmfval 17286* 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 17287* 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 17288* 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 17289* 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 17290* 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 17291* 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 17292* 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 17293* 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 17294 Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcntop2 17295 Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcnptop1 17296 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcnptop2 17297 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremiscnp3 17298* The predicate " is a continuous function from topology to topology at point ." (Contributed by NM, 15-May-2007.)
TopOn TopOn

Theoremcnprcl 17299 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcnf 17300 A continuous function is a mapping. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

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