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

Theoremordtcld1 17201* A downward ray is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtcld2 17202* An upward ray is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtcld3 17203* A closed interval is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordttop 17204 The order topology is a topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtcnv 17205 The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop ordTop

Theoremordtrest 17206 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 17207* Lemma for ordtrest2 17208. (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop

Theoremordtrest2 17208* 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 17209 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop TopOn

Theoremletop 17210 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremletopuni 17211 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremxrstopn 17212 The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
ordTop

Theoremxrstps 17213 The extended real number structure is a topological space. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremleordtvallem1 17214* Lemma for leordtval 17217. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremleordtvallem2 17215* Lemma for leordtval 17217. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremleordtval2 17216 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremleordtval 17217 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremiccordt 17218 A closed interval is closed in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremiocpnfordt 17219 An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremicomnfordt 17220 An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremiooordt 17221 An open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremreordt 17222 The real numbers are an open set in the topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremlecldbas 17223 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 17224* A neighborhood of contains an unbounded interval based at a real number. Together with xrtgioo 18776 (which describes neighborhoods of ) and mnfnei 17225, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 17221 and similar ensure that it has all the sets we want). (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremmnfnei 17225* A neighborhood of contains an unbounded interval based at a real number. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtrestixx 17226* 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 17227 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 17228 Extend class notation with the set of continuous functions between topologies.

Syntaxccnp 17229 Extend class notation with the set of functions between topologies continuous at a point.

Syntaxclm 17230 Extend class notation with a function on topological spaces whose value is the convergence relation for limit sequences in the space.

Definitiondf-cn 17231* 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 17239 for the predicate form. (Contributed by NM, 17-Oct-2006.)

Definitiondf-cnp 17232* 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 17233* 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 5759, and we use the otherwise unnecessary conjunct to ensure that. (Contributed by NM, 7-Sep-2006.)

Theoremlmrel 17234 The topological space convergence relation is a relation. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)

Theoremlmrcl 17235 Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.)

Theoremlmfval 17236* 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 17237* 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 17238* 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 17239* 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 17240* 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 17241* 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 17242* 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 17243* 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 17244 Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcntop2 17245 Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcnptop1 17246 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcnptop2 17247 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremiscnp3 17248* The predicate " is a continuous function from topology to topology at point ." (Contributed by NM, 15-May-2007.)
TopOn TopOn

Theoremcnprcl 17249 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcnf 17250 A continuous function is a mapping. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Theoremcnpf 17251 A continuous function at point is a mapping. (Contributed by FL, 17-Nov-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Theoremcnpcl 17252 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 17253 A continuous function is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremcnpf2 17254 A continuous function at point is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremcnprcl2 17255 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
TopOn

Theoremtgcn 17256* 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 17257* 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 17258* 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 17259 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 17260* Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.)

Theoremidcn 17261 A restricted identity function is a continuous function. (Contributed by FL, 27-Dec-2006.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
TopOn

Theoremlmbr 17262* 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 17233. (Contributed by Mario Carneiro, 14-Nov-2013.)
TopOn

Theoremlmbr2 17263* 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 17264* Express the binary relation "sequence converges to point " in a metric space using an arbitrary set of upper integers. This version of lmbr2 17263 presupposes that is a function. (Contributed by Mario Carneiro, 14-Nov-2013.)
TopOn

Theoremlmconst 17265 A constant sequence converges to its value. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)
TopOn

Theoremlmcvg 17266* Convergence property of a converging sequence. (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremiscnp4 17267* 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 17268* A condition for continuity at a point in terms of neighborhoods. (Contributed by Jeff Hankins, 7-Sep-2009.)

Theoremcnima 17269 An open subset of the codomain of a continuous function has an open preimage. (Contributed by FL, 15-Dec-2006.)

Theoremcnco 17270 The composition of two continuous functions is a continuous function. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Theoremcnpco 17271 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 17272 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 17273* 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 17274 Property of the preimage of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremcnntri 17275 Property of the preimage of an interior. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremcnclsi 17276 Property of the image of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremcncls2 17277* Continuity in terms of closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn TopOn

Theoremcncls 17278* Continuity in terms of closure. (Contributed by Jeff Hankins, 1-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
TopOn TopOn

Theoremcnntr 17279* Continuity in terms of interior. (Contributed by Jeff Hankins, 2-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
TopOn TopOn

Theoremcnss1 17280 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 17281 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 17282 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 17283 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 17284* 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 17285* 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 17286* Continuity in terms of neighborhoods. (Contributed by Thierry Arnoux, 3-Jan-2018.)

Theoremcnconst2 17287 A constant function is continuous. (Contributed by Mario Carneiro, 19-Mar-2015.)
TopOn TopOn

Theoremcnconst 17288 A constant function is continuous. (Contributed by FL, 15-Jan-2007.) (Proof shortened by Mario Carneiro, 19-Mar-2015.)
TopOn TopOn

Theoremcnrest 17289 Continuity of a restriction from a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 21-Aug-2015.)
t

Theoremcnrest2 17290 Equivalence of continuity in the parent topology and continuity in a subspace. (Contributed by Jeff Hankins, 10-Jul-2009.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
TopOn t

Theoremcnrest2r 17291 Equivalence of continuity in the parent topology and continuity in a subspace. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Jun-2014.)
t

Theoremcnpresti 17292 One direction of cnprest 17293 under the weaker condition that the point is in the subset rather than the interior of the subset. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 1-May-2015.)
t

Theoremcnprest 17293 Equivalence of continuity at a point and continuity of the restricted function at a point. (Contributed by Mario Carneiro, 8-Aug-2014.)
t

Theoremcnprest2 17294 Equivalence of point-continuity in the parent topology and point-continuity in a subspace. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 21-Aug-2015.)
t

Theoremcndis 17295 Every function is continuous when the domain is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn

Theoremcnindis 17296 Every function is continuous when the codomain is indiscrete (trivial). (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn

Theoremcnpdis 17297 If is an isolated point in (or equivalently, the singleton is open in ), then every function is continuous at . (Contributed by Mario Carneiro, 9-Sep-2015.)
TopOn TopOn

Theorempaste 17298 Pasting lemma. If and are closed sets in with , then any function whose restrictions to and are continuous is continuous on all of . (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
t        t

Theoremlmfpm 17299 If converges, then is a partial function. (Contributed by Mario Carneiro, 23-Dec-2013.)
TopOn

Theoremlmfss 17300 Inclusion of a function having a limit (used to ensure the limit relation is a set, under our definition). (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
TopOn

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