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Theorem List for Metamath Proof Explorer - 21101-21200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcnf2 21101 A continuous function is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)

Theoremcnpf2 21102 A continuous function at point 𝑃 is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋𝑌)

Theoremcnprcl2 21103 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃𝑋)

Theoremtgcn 21104* The continuity 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.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 = (topGen‘𝐵))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))       (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐵 (𝐹𝑦) ∈ 𝐽)))

Theoremtgcnp 21105* 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.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 = (topGen‘𝐵))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝑃𝑋)       (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐵 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))

Theoremsubbascn 21106* The continuity 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.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐵𝑉)    &   (𝜑𝐾 = (topGen‘(fi‘𝐵)))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))       (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐵 (𝐹𝑦) ∈ 𝐽)))

Theoremssidcn 21107 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.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ 𝐾𝐽))

Theoremcnpimaex 21108* Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.)
((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴𝐾 ∧ (𝐹𝑃) ∈ 𝐴) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝐴))

Theoremidcn 21109 A restricted identity function is a continuous function. (Contributed by FL, 27-Dec-2006.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
(𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))

Theoremlmbr 21110* Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a topological space. Definition 1.4-1 of [Kreyszig] p. 25. The condition 𝐹 ⊆ (ℂ × 𝑋) allows us to use objects more general than sequences when convenient; see the comment in df-lm 21081. (Contributed by Mario Carneiro, 14-Nov-2013.)
(𝜑𝐽 ∈ (TopOn‘𝑋))       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢))))

Theoremlmbr2 21111* Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space using an arbitrary upper set of integers. (Contributed by Mario Carneiro, 14-Nov-2013.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))))

Theoremlmbrf 21112* Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space using an arbitrary upper set of integers. This version of lmbr2 21111 presupposes that 𝐹 is a function. (Contributed by Mario Carneiro, 14-Nov-2013.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍𝑋)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝐴𝑢))))

Theoremlmconst 21113 A constant sequence converges to its value. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)
𝑍 = (ℤ𝑀)       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑍 × {𝑃})(⇝𝑡𝐽)𝑃)

Theoremlmcvg 21114* Convergence property of a converging sequence. (Contributed by Mario Carneiro, 14-Nov-2013.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑃𝑈)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹(⇝𝑡𝐽)𝑃)    &   (𝜑𝑈𝐽)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑈)

Theoremiscnp4 21115* The predicate "𝐹 is a continuous function from topology 𝐽 to topology 𝐾 at point 𝑃." in terms of neighborhoods. (Contributed by FL, 18-Jul-2011.) (Revised by Mario Carneiro, 10-Sep-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹𝑥) ⊆ 𝑦)))

Theoremcnpnei 21116* A condition for continuity at a point in terms of neighborhoods. (Contributed by Jeff Hankins, 7-Sep-2009.)
𝑋 = 𝐽    &   𝑌 = 𝐾       (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋𝑌) ∧ 𝐴𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝐴)})(𝐹𝑦) ∈ ((nei‘𝐽)‘{𝐴})))

Theoremcnima 21117 An open subset of the codomain of a continuous function has an open preimage. (Contributed by FL, 15-Dec-2006.)
((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝐾) → (𝐹𝐴) ∈ 𝐽)

Theoremcnco 21118 The composition of two continuous functions is a continuous function. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐺𝐹) ∈ (𝐽 Cn 𝐿))

Theoremcnpco 21119 The composition of two continuous functions at point 𝑃 is a continuous function at point 𝑃. Proposition of [BourbakiTop1] p. I.9. (Contributed by FL, 16-Nov-2006.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → (𝐺𝐹) ∈ ((𝐽 CnP 𝐿)‘𝑃))

Theoremcnclima 21120 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.)
((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (𝐹𝐴) ∈ (Clsd‘𝐽))

Theoremiscncl 21121* 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.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))))

Theoremcncls2i 21122 Property of the preimage of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝑌 = 𝐾       ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑌) → ((cls‘𝐽)‘(𝐹𝑆)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑆)))

Theoremcnntri 21123 Property of the preimage of an interior. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝑌 = 𝐾       ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑌) → (𝐹 “ ((int‘𝐾)‘𝑆)) ⊆ ((int‘𝐽)‘(𝐹𝑆)))

Theoremcnclsi 21124 Property of the image of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝑋 = 𝐽       ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → (𝐹 “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐾)‘(𝐹𝑆)))

Theoremcncls2 21125* Continuity in terms of closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥)))))

Theoremcncls 21126* Continuity in terms of closure. (Contributed by Jeff Hankins, 1-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥)))))

Theoremcnntr 21127* Continuity in terms of interior. (Contributed by Jeff Hankins, 2-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌(𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥)))))

Theoremcnss1 21128 If the topology 𝐾 is finer than 𝐽, then there are more continuous functions from 𝐾 than from 𝐽. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
𝑋 = 𝐽       ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝐽 Cn 𝐿) ⊆ (𝐾 Cn 𝐿))

Theoremcnss2 21129 If the topology 𝐾 is finer than 𝐽, then there are fewer continuous functions into 𝐾 than into 𝐽 from some other space. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
𝑌 = 𝐾       ((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn 𝐿))

Theoremcncnpi 21130 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.)
𝑋 = 𝐽       ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))

Theoremcnsscnp 21131 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.)
𝑋 = 𝐽       (𝑃𝑋 → (𝐽 Cn 𝐾) ⊆ ((𝐽 CnP 𝐾)‘𝑃))

Theoremcncnp 21132* 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.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))

Theoremcncnp2 21133* 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.)
𝑋 = 𝐽    &   𝑌 = 𝐾       (𝑋 ≠ ∅ → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))

Theoremcnnei 21134* Continuity in terms of neighborhoods. (Contributed by Thierry Arnoux, 3-Jan-2018.)
𝑋 = 𝐽    &   𝑌 = 𝐾       ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑝𝑋𝑤 ∈ ((nei‘𝐾)‘{(𝐹𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹𝑣) ⊆ 𝑤))

Theoremcnconst2 21135 A constant function is continuous. (Contributed by Mario Carneiro, 19-Mar-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) → (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾))

Theoremcnconst 21136 A constant function is continuous. (Contributed by FL, 15-Jan-2007.) (Proof shortened by Mario Carneiro, 19-Mar-2015.)
(((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐵𝑌𝐹:𝑋⟶{𝐵})) → 𝐹 ∈ (𝐽 Cn 𝐾))

Theoremcnrest 21137 Continuity of a restriction from a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 21-Aug-2015.)
𝑋 = 𝐽       ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾))

Theoremcnrest2 21138 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.)
((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾t 𝐵))))

Theoremcnrest2r 21139 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.)
(𝐾 ∈ Top → (𝐽 Cn (𝐾t 𝐵)) ⊆ (𝐽 Cn 𝐾))

Theoremcnpresti 21140 One direction of cnprest 21141 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.)
𝑋 = 𝐽       ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃))

Theoremcnprest 21141 Equivalence of continuity at a point and continuity of the restricted function at a point. (Contributed by Mario Carneiro, 8-Aug-2014.)
𝑋 = 𝐽    &   𝑌 = 𝐾       (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃)))

Theoremcnprest2 21142 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.)
𝑋 = 𝐽    &   𝑌 = 𝐾       ((𝐾 ∈ Top ∧ 𝐹:𝑋𝐵𝐵𝑌) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃)))

Theoremcndis 21143 Every function is continuous when the domain is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝒫 𝐴 Cn 𝐽) = (𝑋𝑚 𝐴))

Theoremcnindis 21144 Every function is continuous when the codomain is indiscrete (trivial). (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝐽 Cn {∅, 𝐴}) = (𝐴𝑚 𝑋))

Theoremcnpdis 21145 If 𝐴 is an isolated point in 𝑋 (or equivalently, the singleton {𝐴} is open in 𝑋), then every function is continuous at 𝐴. (Contributed by Mario Carneiro, 9-Sep-2015.)
(((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → ((𝐽 CnP 𝐾)‘𝐴) = (𝑌𝑚 𝑋))

Theorempaste 21146 Pasting lemma. If 𝐴 and 𝐵 are closed sets in 𝑋 with 𝐴𝐵 = 𝑋, then any function whose restrictions to 𝐴 and 𝐵 are continuous is continuous on all of 𝑋. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
𝑋 = 𝐽    &   𝑌 = 𝐾    &   (𝜑𝐴 ∈ (Clsd‘𝐽))    &   (𝜑𝐵 ∈ (Clsd‘𝐽))    &   (𝜑 → (𝐴𝐵) = 𝑋)    &   (𝜑𝐹:𝑋𝑌)    &   (𝜑 → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾))    &   (𝜑 → (𝐹𝐵) ∈ ((𝐽t 𝐵) Cn 𝐾))       (𝜑𝐹 ∈ (𝐽 Cn 𝐾))

Theoremlmfpm 21147 If 𝐹 converges, then 𝐹 is a partial function. (Contributed by Mario Carneiro, 23-Dec-2013.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝑃) → 𝐹 ∈ (𝑋pm ℂ))

Theoremlmfss 21148 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.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝑃) → 𝐹 ⊆ (ℂ × 𝑋))

Theoremlmcl 21149 Closure of a limit. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝑃) → 𝑃𝑋)

Theoremlmss 21150 Limit on a subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)
𝐾 = (𝐽t 𝑌)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑌𝑉)    &   (𝜑𝐽 ∈ Top)    &   (𝜑𝑃𝑌)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍𝑌)       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃𝐹(⇝𝑡𝐾)𝑃))

Theoremsslm 21151 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.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (⇝𝑡𝐾) ⊆ (⇝𝑡𝐽))

Theoremlmres 21152 A function converges iff its restriction to an upper integers set converges. (Contributed by Mario Carneiro, 31-Dec-2013.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝑋pm ℂ))    &   (𝜑𝑀 ∈ ℤ)       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ↾ (ℤ𝑀))(⇝𝑡𝐽)𝑃))

Theoremlmff 21153* If 𝐹 converges, there is some upper integer set on which 𝐹 is a total function. (Contributed by Mario Carneiro, 31-Dec-2013.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹 ∈ dom (⇝𝑡𝐽))       (𝜑 → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋)

Theoremlmcls 21154* 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.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹(⇝𝑡𝐽)𝑃)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ 𝑆)    &   (𝜑𝑆𝑋)       (𝜑𝑃 ∈ ((cls‘𝐽)‘𝑆))

Theoremlmcld 21155* 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.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹(⇝𝑡𝐽)𝑃)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ 𝑆)    &   (𝜑𝑆 ∈ (Clsd‘𝐽))       (𝜑𝑃𝑆)

Theoremlmcnp 21156 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.)
(𝜑𝐹(⇝𝑡𝐽)𝑃)    &   (𝜑𝐺 ∈ ((𝐽 CnP 𝐾)‘𝑃))       (𝜑 → (𝐺𝐹)(⇝𝑡𝐾)(𝐺𝑃))

Theoremlmcn 21157 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.)
(𝜑𝐹(⇝𝑡𝐽)𝑃)    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝐺𝐹)(⇝𝑡𝐾)(𝐺𝑃))

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

Syntaxct0 21158 Extend class notation with the class of all T0 spaces.
class Kol2

Syntaxct1 21159 Extend class notation to include T1 spaces (also called Fréchet spaces).
class Fre

Syntaxcha 21160 Extend class notation with the class of all Hausdorff spaces.
class Haus

Syntaxcreg 21161 Extend class notation with the class of all regular topologies.
class Reg

Syntaxcnrm 21162 Extend class notation with the class of all normal topologies.
class Nrm

Syntaxccnrm 21163 Extend class notation with the class of all completely normal topologies.
class CNrm

Syntaxcpnrm 21164 Extend class notation with the class of all perfectly normal topologies.
class PNrm

Definitiondf-t0 21165* Define T0 or Kolmogorov spaces. A T0 space satisfies a kind of "topological extensionality" principle (compare ax-ext 2631): 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 21199) 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.)
Kol2 = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 𝑗(∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)}

Definitiondf-t1 21166* The class of all T1 spaces, also called Fréchet spaces. Morris, Topology without tears, p. 30 ex. 3. (Contributed by FL, 18-Jun-2007.)
Fre = {𝑥 ∈ Top ∣ ∀𝑎 𝑥{𝑎} ∈ (Clsd‘𝑥)}

Definitiondf-haus 21167* 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.)
Haus = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 𝑗(𝑥𝑦 → ∃𝑛𝑗𝑚𝑗 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))}

Definitiondf-reg 21168* 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.)
Reg = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑧𝑗 (𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}

Definitiondf-nrm 21169* Define normal spaces. A space is normal if disjoint closed sets can be separated by neighborhoods. (Contributed by Jeff Hankins, 1-Feb-2010.)
Nrm = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦 ∈ ((Clsd‘𝑗) ∩ 𝒫 𝑥)∃𝑧𝑗 (𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}

Definitiondf-cnrm 21170* Define completely normal spaces. A space is completely normal if all its subspaces are normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
CNrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝒫 𝑗(𝑗t 𝑥) ∈ Nrm}

Definitiondf-pnrm 21171* 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 = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗𝑚 ℕ) ↦ ran 𝑓)}

Theoremist0 21172* 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 21197. (Contributed by Jeff Hankins, 1-Feb-2010.)
𝑋 = 𝐽       (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))

Theoremist1 21173* The predicate 𝐽 is T1. (Contributed by FL, 18-Jun-2007.)
𝑋 = 𝐽       (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑎𝑋 {𝑎} ∈ (Clsd‘𝐽)))

Theoremishaus 21174* Express the predicate "𝐽 is a Hausdorff space." (Contributed by NM, 8-Mar-2007.)
𝑋 = 𝐽       (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))))

Theoremiscnrm 21175* The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝑋 = 𝐽       (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐽t 𝑥) ∈ Nrm))

Theoremt0sep 21176* Any two topologically indistinguishable points in a T0 space are identical. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Kol2 ∧ (𝐴𝑋𝐵𝑋)) → (∀𝑥𝐽 (𝐴𝑥𝐵𝑥) → 𝐴 = 𝐵))

Theoremt0dist 21177* Any two distinct points in a T0 space are topologically distinguishable. (Contributed by Jeff Hankins, 1-Feb-2010.)
𝑋 = 𝐽       ((𝐽 ∈ Kol2 ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → ∃𝑜𝐽 ¬ (𝐴𝑜𝐵𝑜))

Theoremt1sncld 21178 In a T1 space, one-point sets are closed. (Contributed by Jeff Hankins, 1-Feb-2010.)
𝑋 = 𝐽       ((𝐽 ∈ Fre ∧ 𝐴𝑋) → {𝐴} ∈ (Clsd‘𝐽))

Theoremt1ficld 21179 In a T1 space, finite sets are closed. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝑋 = 𝐽       ((𝐽 ∈ Fre ∧ 𝐴𝑋𝐴 ∈ Fin) → 𝐴 ∈ (Clsd‘𝐽))

Theoremhausnei 21180* Neighborhood property of a Hausdorff space. (Contributed by NM, 8-Mar-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Haus ∧ (𝑃𝑋𝑄𝑋𝑃𝑄)) → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅))

Theoremt0top 21181 A T0 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
(𝐽 ∈ Kol2 → 𝐽 ∈ Top)

Theoremt1top 21182 A T1 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
(𝐽 ∈ Fre → 𝐽 ∈ Top)

Theoremhaustop 21183 A Hausdorff space is a topology. (Contributed by NM, 5-Mar-2007.)
(𝐽 ∈ Haus → 𝐽 ∈ Top)

Theoremisreg 21184* 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.)
(𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))

Theoremregtop 21185 A regular space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
(𝐽 ∈ Reg → 𝐽 ∈ Top)

Theoremregsep 21186* In a regular space, every neighborhood of a point contains a closed subneighborhood. (Contributed by Mario Carneiro, 25-Aug-2015.)
((𝐽 ∈ Reg ∧ 𝑈𝐽𝐴𝑈) → ∃𝑥𝐽 (𝐴𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈))

Theoremisnrm 21187* 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.)
(𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))

Theoremnrmtop 21188 A normal space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
(𝐽 ∈ Nrm → 𝐽 ∈ Top)

Theoremcnrmtop 21189 A completely normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝐽 ∈ CNrm → 𝐽 ∈ Top)

Theoremiscnrm2 21190* The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 𝑋(𝐽t 𝑥) ∈ Nrm))

Theoremispnrm 21191* The property of being perfectly normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓)))

Theorempnrmnrm 21192 A perfectly normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝐽 ∈ PNrm → 𝐽 ∈ Nrm)

Theorempnrmtop 21193 A perfectly normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝐽 ∈ PNrm → 𝐽 ∈ Top)

Theorempnrmcld 21194* A closed set in a perfectly normal space is a countable intersection of open sets. (Contributed by Mario Carneiro, 26-Aug-2015.)
((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → ∃𝑓 ∈ (𝐽𝑚 ℕ)𝐴 = ran 𝑓)

Theorempnrmopn 21195* An open set in a perfectly normal space is a countable union of closed sets. (Contributed by Mario Carneiro, 26-Aug-2015.)
((𝐽 ∈ PNrm ∧ 𝐴𝐽) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑𝑚 ℕ)𝐴 = ran 𝑓)

Theoremist0-2 21196* The predicate "is a T0 space". (Contributed by Mario Carneiro, 24-Aug-2015.)
(𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))

Theoremist0-3 21197* The predicate "is a T0 space," expressed in more familiar terms. (Contributed by Jeff Hankins, 1-Feb-2010.)
(𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑜𝐽 ((𝑥𝑜 ∧ ¬ 𝑦𝑜) ∨ (¬ 𝑥𝑜𝑦𝑜)))))

Theoremcnt0 21198 The preimage of a T0 topology under an injective map is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
((𝐾 ∈ Kol2 ∧ 𝐹:𝑋1-1𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Kol2)

Theoremist1-2 21199* An alternate characterization of T1 spaces. (Contributed by Jeff Hankins, 31-Jan-2010.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
(𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))

Theoremt1t0 21200 A T1 space is a T0 space. (Contributed by Jeff Hankins, 1-Feb-2010.)
(𝐽 ∈ Fre → 𝐽 ∈ Kol2)

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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
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