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Theorem List for Metamath Proof Explorer - 17301-17400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcnconst2 17301 A constant function is continuous. (Contributed by Mario Carneiro, 19-Mar-2015.)
TopOn TopOn

Theoremcnconst 17302 A constant function is continuous. (Contributed by FL, 15-Jan-2007.) (Proof shortened by Mario Carneiro, 19-Mar-2015.)
TopOn TopOn

Theoremcnrest 17303 Continuity of a restriction from a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 21-Aug-2015.)
t

Theoremcnrest2 17304 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 17305 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 17306 One direction of cnprest 17307 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 17307 Equivalence of continuity at a point and continuity of the restricted function at a point. (Contributed by Mario Carneiro, 8-Aug-2014.)
t

Theoremcnprest2 17308 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 17309 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 17310 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 17311 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 17312 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 17313 If converges, then is a partial function. (Contributed by Mario Carneiro, 23-Dec-2013.)
TopOn

Theoremlmfss 17314 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

Theoremlmcl 17315 Closure of a limit. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
TopOn

Theoremlmss 17316 Limit on a subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)
t

Theoremsslm 17317 A finer topology has fewer convergent sequences (but the sequences that do converge, converge to the same value). (Contributed by Mario Carneiro, 15-Sep-2015.)
TopOn TopOn

Theoremlmres 17318 A function converges iff its restriction to an upper integers set converges. (Contributed by Mario Carneiro, 31-Dec-2013.)
TopOn

Theoremlmff 17319* If converges, there is some upper integer set on which is a total function. (Contributed by Mario Carneiro, 31-Dec-2013.)
TopOn

Theoremlmcls 17320* Any convergent sequence of points in a subset of a topological space converges to a point in the closure of the subset. (Contributed by Mario Carneiro, 30-Dec-2013.) (Revised by Mario Carneiro, 1-May-2014.)
TopOn

Theoremlmcld 17321* Any convergent sequence of points in a closed subset of a topological space converges to a point in the set. (Contributed by Mario Carneiro, 30-Dec-2013.)
TopOn

Theoremlmcnp 17322 The image of a convergent sequence under a continuous map is convergent to the image of the original point. (Contributed by Mario Carneiro, 3-May-2014.)

Theoremlmcn 17323 The image of a convergent sequence under a continuous map is convergent to the image of the original point. (Contributed by Mario Carneiro, 3-May-2014.)

11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...

Syntaxct0 17324 Extend class notation with the class of all T0 spaces.

Syntaxct1 17325 Extend class notation to include T1 spaces (also called Fréchet spaces).

Syntaxcha 17326 Extend class notation with the class of all Hausdorff spaces.

Syntaxcreg 17327 Extend class notation with the class of all regular topologies.

Syntaxcnrm 17328 Extend class notation with the class of all normal topologies.

Syntaxccnrm 17329 Extend class notation with the class of all completely normal topologies.
CNrm

Syntaxcpnrm 17330 Extend class notation with the class of all perfectly normal topologies.
PNrm

Definitiondf-t0 17331* Define T0 or Kolmogorov spaces. A T0 space satisfies a kind of "topological extensionality" principle (compare ax-ext 2385): any two points which are members of the same open sets are equal, or in contraposition, for any two distinct points there is an open set which contains one point but not the other. This differs from T1 spaces (see ist1-2 17365) in that in a T1 space you can choose which point will be in the open set and which outside; in a T0 space you only know that one of the two points is in the set. (Contributed by Jeff Hankins, 1-Feb-2010.)

Definitiondf-t1 17332* The class of all T1 spaces, also called Fréchet spaces. Morris, Topology without tears, p. 30 ex. 3. (Contributed by FL, 18-Jun-2007.)

Definitiondf-haus 17333* Define the class of all Hausdorff spaces. A Hausdorff space is a topology in which distinct points have disjoint open neighborhoods. Definition of Hausdorff space in [Munkres] p. 98. (Contributed by NM, 8-Mar-2007.)

Definitiondf-reg 17334* Define regular spaces. A space is regular if a point and a closed set can be separated by neighborhoods. (Contributed by Jeff Hankins, 1-Feb-2010.)

Definitiondf-nrm 17335* Define normal spaces. A space is normal if disjoint closed sets can be separated by neighborhoods. (Contributed by Jeff Hankins, 1-Feb-2010.)

Definitiondf-cnrm 17336* Define completely normal spaces. A space is completely normal if all its subspaces are normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
CNrm t

Definitiondf-pnrm 17337* Define perfectly normal spaces. A space is perfectly normal if it is normal and every closed set is a G&delta; set, meaning that it is a countable intersection of open sets. (Contributed by Mario Carneiro, 26-Aug-2015.)
PNrm

Theoremist0 17338* The predicate "is a T0 space." Every pair of distinct points is topologically distinguishable. For the way this definition is usually encountered, see ist0-3 17363. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremist1 17339* The predicate is T1. (Contributed by FL, 18-Jun-2007.)

Theoremishaus 17340* Express the predicate " is a Hausdorff space." (Contributed by NM, 8-Mar-2007.)

Theoremiscnrm 17341* The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
CNrm t

Theoremt0sep 17342* Any two topologically indistinguishable points in a T0 space are identical. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremt0dist 17343* Any two distinct points in a T0 space are topologically distinguishable. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremt1sncld 17344 In a T1 space, one-point sets are closed. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremt1ficld 17345 In a T1 space, finite sets are closed. (Contributed by Mario Carneiro, 25-Dec-2016.)

Theoremhausnei 17346* Neighborhood property of a Hausdorff space. (Contributed by NM, 8-Mar-2007.)

Theoremt0top 17347 A T0 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremt1top 17348 A T1 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremhaustop 17349 A Hausdorff space is a topology. (Contributed by NM, 5-Mar-2007.)

Theoremisreg 17350* The predicate "is a regular space." In a regular space, any open neighborhood has a closed subneighborhood. Note that some authors require the space to be Hausdorff (which would make it the same as T3), but we reserve the phrase "regular Hausdorff" for that as many topologists do. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 25-Aug-2015.)

Theoremregtop 17351 A regular space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremregsep 17352* In a regular space, every neighborhood of a point contains a closed subneighborhood. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremisnrm 17353* The predicate "is a normal space." Much like the case for regular spaces, normal does not imply Hausdorff or even regular. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 24-Aug-2015.)

Theoremnrmtop 17354 A normal space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremcnrmtop 17355 A completely normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
CNrm

Theoremiscnrm2 17356* The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
TopOn CNrm t

Theoremispnrm 17357* The property of being perfectly normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
PNrm

Theorempnrmnrm 17358 A perfectly normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
PNrm

Theorempnrmtop 17359 A perfectly normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
PNrm

Theorempnrmcld 17360* A closed set in a perfectly normal space is a countable intersection of open sets. (Contributed by Mario Carneiro, 26-Aug-2015.)
PNrm

Theorempnrmopn 17361* An open set in a perfectly normal space is a countable union of closed sets. (Contributed by Mario Carneiro, 26-Aug-2015.)
PNrm

Theoremist0-2 17362* The predicate "is a T0 space". (Contributed by Mario Carneiro, 24-Aug-2015.)
TopOn

Theoremist0-3 17363* The predicate "is a T0 space," expressed in more familiar terms. (Contributed by Jeff Hankins, 1-Feb-2010.)
TopOn

Theoremcnt0 17364 The preimage of a T0 topology under an injective map is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremist1-2 17365* An alternate characterization of T1 spaces. (Contributed by Jeff Hankins, 31-Jan-2010.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
TopOn

Theoremt1t0 17366 A T1 space is a T0 space. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremist1-3 17367* A space is T1 iff every point is the only point in the intersection of all open sets containing that point. (Contributed by Jeff Hankins, 31-Jan-2010.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
TopOn

Theoremcnt1 17368 The preimage of a T1 topology under an injective map is T1. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremishaus2 17369* Express the predicate " is a Hausdorff space." (Contributed by NM, 8-Mar-2007.)
TopOn

Theoremhaust1 17370 A Hausdorff space is a T1 space. (Contributed by FL, 11-Jun-2007.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)

Theoremhausnei2 17371* The Hausdorff condition still holds if one considers general neighborhoods instead of open sets. (Contributed by Jeff Hankins, 5-Sep-2009.)
TopOn

Theoremcnhaus 17372 The preimage of a Hausdorff topology under an injective map is Hausdorff. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremnrmsep3 17373* In a normal space, given a closed set inside an open set , there is an open set such that . (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremnrmsep2 17374* In a normal space, any two disjoint closed sets have the property that each one is a subset of an open set whose closure is disjoint from the other. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 24-Aug-2015.)

Theoremnrmsep 17375* In a normal space, disjoint closed sets are separated by open sets. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremisnrm2 17376* An alternate characterization of normality. This is the important property in the proof of Urysohn's lemma. (Contributed by Jeff Hankins, 1-Feb-2010.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)

Theoremisnrm3 17377* A topological space is normal iff any two disjoint closed sets are separated by open sets. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremcnrmi 17378 A subspace of a completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
CNrm t

Theoremcnrmnrm 17379 A completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
CNrm

Theoremrestcnrm 17380 A subspace of a completely normal space is completely normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
CNrm t CNrm

Theoremresthauslem 17381 Lemma for resthaus 17386 and similar theorems. If the topological property is preserved under injective preimages, then property passes to subspaces. (Contributed by Mario Carneiro, 25-Aug-2015.)
t t        t

Theoremlpcls 17382 The limit points of the closure of a subset are the same as the limit points of the set in a T1 space. (Contributed by Mario Carneiro, 26-Dec-2016.)

Theoremperfcls 17383 A subset of a perfect space is perfect iff its closure is perfect (and the closure is an actual perfect set, since it is both closed and perfect in the subspace topology). (Contributed by Mario Carneiro, 26-Dec-2016.)
t Perf t Perf

Theoremrestt0 17384 A subspace of a T0 topology is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
t

Theoremrestt1 17385 A subspace of a T1 topology is T1. (Contributed by Mario Carneiro, 25-Aug-2015.)
t

Theoremresthaus 17386 A subspace of a Hausdorff topology is Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
t

Theoremt1sep2 17387* Any two points in a T1 space which have no separation are equal. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremt1sep 17388* Any two distinct points in a T1 space are separated by an open set. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremsncld 17389 A singleton is closed in a Hausdorff space. (Contributed by NM, 5-Mar-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)

Theoremsshauslem 17390 Lemma for sshaus 17393 and similar theorems. If the topological property is preserved under injective preimages, then a topology finer than one with property also has property . (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremsst0 17391 A topology finer than a T0 topology is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremsst1 17392 A topology finer than a T1 topology is T1. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremsshaus 17393 A topology finer than a Hausdorff topology is Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.)
TopOn

Theoremregsep2 17394* In a regular space, a closed set is separated by open sets from a point not in it. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 25-Aug-2015.)

Theoremisreg2 17395* A topological space is regular if any closed set is separated from any point not in it by neighborhoods. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremdnsconst 17396 If a continuous mapping to a T1 space is constant on a dense subset, it is constant on the entire space. Note that means " is dense in " and means " is constant on " (see funconstss 5807). (Contributed by NM, 15-Mar-2007.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)

Theoremordtt1 17397 The order topology is T1 for any poset. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremlmmo 17398 A sequence in a Hausdorff space converges to at most one limit. Part of Lemma 1.4-2(a) of [Kreyszig] p. 26. (Contributed by NM, 31-Jan-2008.) (Proof shortened by Mario Carneiro, 1-May-2014.)

Theoremlmfun 17399 The convergence relation is function-like in a Hausdorff space. (Contributed by Mario Carneiro, 26-Dec-2013.)

Theoremdishaus 17400 A discrete topology is Hausdorff. Morris, Topology without tears, p.72, ex. 13. (Contributed by FL, 24-Jun-2007.) (Proof shortened by Mario Carneiro, 8-Apr-2015.)

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