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Theorem List for Metamath Proof Explorer - 17001-17100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremssidcn 17001 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.)
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Theoremcnpimaex 17002* Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.)

Theoremidcn 17003 A restricted identity function is a continuous function. (Contributed by FL, 27-Dec-2006.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
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Theoremlmbr 17004* 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 16975. (Contributed by Mario Carneiro, 14-Nov-2013.)
TopOn

Theoremlmbr2 17005* 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 17006* Express the binary relation "sequence converges to point " in a metric space using an arbitrary set of upper integers. This version of lmbr2 17005 presupposes that is a function. (Contributed by Mario Carneiro, 14-Nov-2013.)
TopOn

Theoremlmconst 17007 A constant sequence converges to its value. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)
TopOn

Theoremlmcvg 17008* Convergence property of a converging sequence. (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremcnpnei 17009* A condition for continuity at a point in terms of neighborhoods. (Contributed by Jeff Hankins, 7-Sep-2009.)

Theoremcnima 17010 An open subset of the codomain of a continuous function has an open preimage. (Contributed by FL, 15-Dec-2006.)

Theoremcnco 17011 The composition of two continuous functions is a continuous function. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Theoremcnpco 17012 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 17013 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 17014* 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.)
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Theoremcncls2i 17015 Property of the preimage of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremcnntri 17016 Property of the preimage of an interior. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremcnclsi 17017 Property of the image of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremcncls2 17018* Continuity in terms of closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
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Theoremcncls 17019* Continuity in terms of closure. (Contributed by Jeff Hankins, 1-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
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Theoremcnntr 17020* Continuity in terms of interior. (Contributed by Jeff Hankins, 2-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
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Theoremcnss1 17021 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 17022 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 17023 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 17024 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 17025* 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.)
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Theoremcncnp2 17026* 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.)

Theoremcnconst2 17027 A constant function is continuous. (Contributed by Mario Carneiro, 19-Mar-2015.)
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Theoremcnconst 17028 A constant function is continuous. (Contributed by FL, 15-Jan-2007.) (Proof shortened by Mario Carneiro, 19-Mar-2015.)
TopOn TopOn

Theoremcnrest 17029 Continuity of a restriction from a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 21-Aug-2015.)
t

Theoremcnrest2 17030 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 17031 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 17032 One direction of cnprest 17033 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 17033 Equivalence of continuity at a point and continuity of the restricted function at a point. (Contributed by Mario Carneiro, 8-Aug-2014.)
t

Theoremcnprest2 17034 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 17035 Every function is continuous when the domain is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
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Theoremcnindis 17036 Every function is continuous when the codomain is indiscrete (trivial). (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
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Theoremcnpdis 17037 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.)
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Theorempaste 17038 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 17039 If converges, then is a partial function. (Contributed by Mario Carneiro, 23-Dec-2013.)
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Theoremlmfss 17040 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 17041 Closure of a limit. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
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Theoremlmss 17042 Limit on a subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)
t

Theoremsslm 17043 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.)
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Theoremlmres 17044 A function converges iff its restriction to an upper integers set converges. (Contributed by Mario Carneiro, 31-Dec-2013.)
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Theoremlmff 17045* If converges, there is some upper integer set on which is a total function. (Contributed by Mario Carneiro, 31-Dec-2013.)
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Theoremlmcls 17046* 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.)
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Theoremlmcld 17047* 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 17048 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 17049 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 17050 Extend class notation with the class of all T0 spaces.

Syntaxct1 17051 Extend class notation to include T1 spaces (also called Fréchet spaces).

Syntaxcha 17052 Extend class notation with the class of all Hausdorff spaces.

Syntaxcreg 17053 Extend class notation with the class of all regular topologies.

Syntaxcnrm 17054 Extend class notation with the class of all normal topologies.

Syntaxccnrm 17055 Extend class notation with the class of all completely normal topologies.
CNrm

Syntaxcpnrm 17056 Extend class notation with the class of all perfectly normal topologies.
PNrm

Definitiondf-t0 17057* Define T0 or Kolmogorov spaces. A T0 space satisfies a kind of "topological extensionality" principle (compare ax-ext 2277): 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 17091) 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 17058* 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 17059* 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 17060* 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 17061* 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 17062* 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 17063* 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 17064* 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 17089. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremist1 17065* The predicate is T1. (Contributed by FL, 18-Jun-2007.)

Theoremishaus 17066* Express the predicate " is a Hausdorff space." (Contributed by NM, 8-Mar-2007.)

Theoremiscnrm 17067* The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
CNrm t

Theoremt0sep 17068* Any two topologically indistinguishable points in a T0 space are identical. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremt0dist 17069* Any two distinct points in a T0 space are topologically distinguishable. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremt1sncld 17070 In a T1 space, one-point sets are closed. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremt1ficld 17071 In a T1 space, finite sets are closed. (Contributed by Mario Carneiro, 25-Dec-2016.)

Theoremhausnei 17072* Neighborhood property of a Hausdorff space. (Contributed by NM, 8-Mar-2007.)

Theoremt0top 17073 A T0 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremt1top 17074 A T1 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremhaustop 17075 A Hausdorff space is a topology. (Contributed by NM, 5-Mar-2007.)

Theoremisreg 17076* 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 17077 A regular space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremregsep 17078* In a regular space, every neighborhood of a point contains a closed subneighborhood. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremisnrm 17079* 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 17080 A normal space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremcnrmtop 17081 A completely normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
CNrm

Theoremiscnrm2 17082* The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
TopOn CNrm t

Theoremispnrm 17083* The property of being perfectly normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
PNrm

Theorempnrmnrm 17084 A perfectly normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
PNrm

Theorempnrmtop 17085 A perfectly normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
PNrm

Theorempnrmcld 17086* A closed set in a perfectly normal space is a countable intersection of open sets. (Contributed by Mario Carneiro, 26-Aug-2015.)
PNrm

Theorempnrmopn 17087* 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 17088* The predicate "is a T0 space". (Contributed by Mario Carneiro, 24-Aug-2015.)
TopOn

Theoremist0-3 17089* The predicate "is a T0 space," expressed in more familiar terms. (Contributed by Jeff Hankins, 1-Feb-2010.)
TopOn

Theoremcnt0 17090 The preimage of a T0 topology under an injective map is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremist1-2 17091* An alternate characterization of T1 spaces. (Contributed by Jeff Hankins, 31-Jan-2010.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
TopOn

Theoremt1t0 17092 A T1 space is a T0 space. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremist1-3 17093* 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 17094 The preimage of a T1 topology under an injective map is T1. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremishaus2 17095* Express the predicate " is a Hausdorff space." (Contributed by NM, 8-Mar-2007.)
TopOn

Theoremhaust1 17096 A Hausdorff space is a T1 space. (Contributed by FL, 11-Jun-2007.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)

Theoremhausnei2 17097* The Hausdorff condition still holds if one considers general neighborhoods instead of open sets. (Contributed by Jeff Hankins, 5-Sep-2009.)
TopOn

Theoremcnhaus 17098 The preimage of a Hausdorff topology under an injective map is Hausdorff. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremnrmsep3 17099* 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 17100* 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.)

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