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Type | Label | Description |
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Statement | ||
Theorem | cnpval 23201* | The set of all functions from topology 𝐽 to topology 𝐾 that are continuous at a point 𝑃. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → ((𝐽 CnP 𝐾)‘𝑃) = {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))}) | ||
Theorem | iscnp 23202* | The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾 at point 𝑃". Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))))) | ||
Theorem | iscn2 23203* | The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾". Definition of continuous function in [Munkres] p. 102. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))) | ||
Theorem | iscnp2 23204* | The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾 at point 𝑃". Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))))) | ||
Theorem | cntop1 23205 | Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | ||
Theorem | cntop2 23206 | Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | ||
Theorem | cnptop1 23207 | Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top) | ||
Theorem | cnptop2 23208 | Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top) | ||
Theorem | iscnp3 23209* | The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾 at point 𝑃". (Contributed by NM, 15-May-2007.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦)))))) | ||
Theorem | cnprcl 23210 | Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 ∈ 𝑋) | ||
Theorem | cnf 23211 | A continuous function is a mapping. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶𝑌) | ||
Theorem | cnpf 23212 | A continuous function at point 𝑃 is a mapping. (Contributed by FL, 17-Nov-2006.) (Revised by Mario Carneiro, 21-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:𝑋⟶𝑌) | ||
Theorem | cnpcl 23213 | The value of a continuous function from 𝐽 to 𝐾 at point 𝑃 belongs to the underlying set of topology 𝐾. (Contributed by FL, 27-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ 𝑌) | ||
Theorem | cnf2 23214 | A continuous function is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶𝑌) | ||
Theorem | cnpf2 23215 | 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:𝑋⟶𝑌) | ||
Theorem | cnprcl2 23216 | Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ 𝑋) | ||
Theorem | tgcn 23217* | 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.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 = (topGen‘𝐵)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽))) | ||
Theorem | tgcnp 23218* | 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.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 = (topGen‘𝐵)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝑃 ∈ 𝑋) ⇒ ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))))) | ||
Theorem | subbascn 23219* | 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.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐾 = (topGen‘(fi‘𝐵))) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽))) | ||
Theorem | ssidcn 23220 | 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) ↔ 𝐾 ⊆ 𝐽)) | ||
Theorem | cnpimaex 23221* | Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.) |
⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝐴) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝐴)) | ||
Theorem | idcn 23222 | 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)) | ||
Theorem | lmbr 23223* | 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 to use objects more general than sequences when convenient; see the comment in df-lm 23194. (Contributed by Mario Carneiro, 14-Nov-2013.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝐹 ↾ 𝑦):𝑦⟶𝑢)))) | ||
Theorem | lmbr2 23224* | 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.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))) | ||
Theorem | lmbrf 23225* | 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 23224 presupposes that 𝐹 is a function. (Contributed by Mario Carneiro, 14-Nov-2013.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶𝑋) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝐴 ∈ 𝑢)))) | ||
Theorem | lmconst 23226 | A constant sequence converges to its value. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.) |
⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → (𝑍 × {𝑃})(⇝𝑡‘𝐽)𝑃) | ||
Theorem | lmcvg 23227* | Convergence property of a converging sequence. (Contributed by Mario Carneiro, 14-Nov-2013.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑃 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃) & ⊢ (𝜑 → 𝑈 ∈ 𝐽) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑈) | ||
Theorem | iscnp4 23228* | The predicate "the class 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) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹 “ 𝑥) ⊆ 𝑦))) | ||
Theorem | cnpnei 23229* | 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:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}))) | ||
Theorem | cnima 23230 | An open subset of the codomain of a continuous function has an open preimage. (Contributed by FL, 15-Dec-2006.) |
⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝐾) → (◡𝐹 “ 𝐴) ∈ 𝐽) | ||
Theorem | cnco 23231 | 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)) → (𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐿)) | ||
Theorem | cnpco 23232 | The composition of a function 𝐹 continuous at 𝑃 with a function continuous at (𝐹‘𝑃) is continuous at 𝑃. Proposition 2 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)‘(𝐹‘𝑃))) → (𝐺 ∘ 𝐹) ∈ ((𝐽 CnP 𝐿)‘𝑃)) | ||
Theorem | cnclima 23233 | 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)) → (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽)) | ||
Theorem | iscncl 23234* | A characterization of a continuity 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:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽)))) | ||
Theorem | cncls2i 23235 | Property of the preimage of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐽)‘(◡𝐹 “ 𝑆)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑆))) | ||
Theorem | cnntri 23236 | Property of the preimage of an interior. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → (◡𝐹 “ ((int‘𝐾)‘𝑆)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑆))) | ||
Theorem | cnclsi 23237 | Property of the image of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → (𝐹 “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑆))) | ||
Theorem | cncls2 23238* | Continuity in terms of closure. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑥))))) | ||
Theorem | cncls 23239* | 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:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥))))) | ||
Theorem | cnntr 23240* | 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:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌(◡𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥))))) | ||
Theorem | cnss1 23241 | 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) ∧ 𝐽 ⊆ 𝐾) → (𝐽 Cn 𝐿) ⊆ (𝐾 Cn 𝐿)) | ||
Theorem | cnss2 23242 | 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) ∧ 𝐿 ⊆ 𝐾) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn 𝐿)) | ||
Theorem | cncnpi 23243 | 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) ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) | ||
Theorem | cnsscnp 23244 | 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 ⇒ ⊢ (𝑃 ∈ 𝑋 → (𝐽 Cn 𝐾) ⊆ ((𝐽 CnP 𝐾)‘𝑃)) | ||
Theorem | cncnp 23245* | 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:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))) | ||
Theorem | cncnp2 23246* | 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) ↔ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) | ||
Theorem | cnnei 23247* | Continuity in terms of neighborhoods. (Contributed by Thierry Arnoux, 3-Jan-2018.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑝 ∈ 𝑋 ∀𝑤 ∈ ((nei‘𝐾)‘{(𝐹‘𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹 “ 𝑣) ⊆ 𝑤)) | ||
Theorem | cnconst2 23248 | A constant function is continuous. (Contributed by Mario Carneiro, 19-Mar-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) → (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾)) | ||
Theorem | cnconst 23249 | 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)) ∧ (𝐵 ∈ 𝑌 ∧ 𝐹:𝑋⟶{𝐵})) → 𝐹 ∈ (𝐽 Cn 𝐾)) | ||
Theorem | cnrest 23250 | 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) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾)) | ||
Theorem | cnrest2 23251 | 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 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵)))) | ||
Theorem | cnrest2r 23252 | 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 (𝐾 ↾t 𝐵)) ⊆ (𝐽 Cn 𝐾)) | ||
Theorem | cnpresti 23253 | One direction of cnprest 23254 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 ⇒ ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃)) | ||
Theorem | cnprest 23254 | 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 ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃))) | ||
Theorem | cnprest2 23255 | 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:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃))) | ||
Theorem | cndis 23256 | Every function is continuous when the domain is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝒫 𝐴 Cn 𝐽) = (𝑋 ↑m 𝐴)) | ||
Theorem | cnindis 23257 | 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) ∧ 𝐴 ∈ 𝑉) → (𝐽 Cn {∅, 𝐴}) = (𝐴 ↑m 𝑋)) | ||
Theorem | cnpdis 23258 | 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) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → ((𝐽 CnP 𝐾)‘𝐴) = (𝑌 ↑m 𝑋)) | ||
Theorem | paste 23259 | Pasting lemma. If 𝐴 and 𝐵 are closed sets in 𝑋 with 𝐴 ∪ 𝐵 = 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 & ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽)) & ⊢ (𝜑 → 𝐵 ∈ (Clsd‘𝐽)) & ⊢ (𝜑 → (𝐴 ∪ 𝐵) = 𝑋) & ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) & ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾)) & ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ ((𝐽 ↾t 𝐵) Cn 𝐾)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | ||
Theorem | lmfpm 23260 | If 𝐹 converges, then 𝐹 is a partial function. (Contributed by Mario Carneiro, 23-Dec-2013.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝐹 ∈ (𝑋 ↑pm ℂ)) | ||
Theorem | lmfss 23261 | 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(⇝𝑡‘𝐽)𝑃) → 𝐹 ⊆ (ℂ × 𝑋)) | ||
Theorem | lmcl 23262 | Closure of a limit. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝑃 ∈ 𝑋) | ||
Theorem | lmss 23263 | Limit on a subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.) |
⊢ 𝐾 = (𝐽 ↾t 𝑌) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝑃 ∈ 𝑌) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶𝑌) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹(⇝𝑡‘𝐾)𝑃)) | ||
Theorem | sslm 23264 | 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) ∧ 𝐽 ⊆ 𝐾) → (⇝𝑡‘𝐾) ⊆ (⇝𝑡‘𝐽)) | ||
Theorem | lmres 23265 | A function converges iff its restriction to an upper integers set converges. (Contributed by Mario Carneiro, 31-Dec-2013.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐹 ∈ (𝑋 ↑pm ℂ)) & ⊢ (𝜑 → 𝑀 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ↾ (ℤ≥‘𝑀))(⇝𝑡‘𝐽)𝑃)) | ||
Theorem | lmff 23266* | If 𝐹 converges, there is some upper integer set on which 𝐹 is a total function. (Contributed by Mario Carneiro, 31-Dec-2013.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ∈ dom (⇝𝑡‘𝐽)) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋) | ||
Theorem | lmcls 23267* | 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 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ 𝑆) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) ⇒ ⊢ (𝜑 → 𝑃 ∈ ((cls‘𝐽)‘𝑆)) | ||
Theorem | lmcld 23268* | 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 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ 𝑆) & ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽)) ⇒ ⊢ (𝜑 → 𝑃 ∈ 𝑆) | ||
Theorem | lmcnp 23269 | 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.) |
⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃) & ⊢ (𝜑 → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ⇒ ⊢ (𝜑 → (𝐺 ∘ 𝐹)(⇝𝑡‘𝐾)(𝐺‘𝑃)) | ||
Theorem | lmcn 23270 | 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.) |
⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃) & ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → (𝐺 ∘ 𝐹)(⇝𝑡‘𝐾)(𝐺‘𝑃)) | ||
Syntax | ct0 23271 | Extend class notation with the class of all T0 spaces. |
class Kol2 | ||
Syntax | ct1 23272 | Extend class notation to include T1 spaces (also called Fréchet spaces). |
class Fre | ||
Syntax | cha 23273 | Extend class notation with the class of all Hausdorff spaces. |
class Haus | ||
Syntax | creg 23274 | Extend class notation with the class of all regular topologies. |
class Reg | ||
Syntax | cnrm 23275 | Extend class notation with the class of all normal topologies. |
class Nrm | ||
Syntax | ccnrm 23276 | Extend class notation with the class of all completely normal topologies. |
class CNrm | ||
Syntax | cpnrm 23277 | Extend class notation with the class of all perfectly normal topologies. |
class PNrm | ||
Definition | df-t0 23278* | Define T0 or Kolmogorov spaces. A T0 space satisfies a kind of "topological extensionality" principle (compare ax-ext 2696): 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 23312) 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 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)} | ||
Definition | df-t1 23279* | 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)} | ||
Definition | df-haus 23280* | Define the class of all Hausdorff (or T2) 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 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝑗 ∃𝑚 ∈ 𝑗 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))} | ||
Definition | df-reg 23281* | 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 ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑗 (𝑦 ∈ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)} | ||
Definition | df-nrm 23282* | 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 ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ ((Clsd‘𝑗) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝑗 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)} | ||
Definition | df-cnrm 23283* | 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(𝑗 ↾t 𝑥) ∈ Nrm} | ||
Definition | df-pnrm 23284* | Define perfectly normal spaces. A space is perfectly normal if it is normal and every closed set is a Gδ 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 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓)} | ||
Theorem | ist0 23285* | 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 23310. (Contributed by Jeff Hankins, 1-Feb-2010.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))) | ||
Theorem | ist1 23286* | The predicate "is a T1 space". (Contributed by FL, 18-Jun-2007.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝑋 {𝑎} ∈ (Clsd‘𝐽))) | ||
Theorem | ishaus 23287* | The predicate "is a Hausdorff space". (Contributed by NM, 8-Mar-2007.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))) | ||
Theorem | iscnrm 23288* | The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐽 ↾t 𝑥) ∈ Nrm)) | ||
Theorem | t0sep 23289* | Any two topologically indistinguishable points in a T0 space are identical. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Kol2 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ↔ 𝐵 ∈ 𝑥) → 𝐴 = 𝐵)) | ||
Theorem | t0dist 23290* | Any two distinct points in a T0 space are topologically distinguishable. (Contributed by Jeff Hankins, 1-Feb-2010.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Kol2 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ≠ 𝐵)) → ∃𝑜 ∈ 𝐽 ¬ (𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜)) | ||
Theorem | t1sncld 23291 | In a T1 space, singletons are closed. (Contributed by Jeff Hankins, 1-Feb-2010.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝐴} ∈ (Clsd‘𝐽)) | ||
Theorem | t1ficld 23292 | In a T1 space, finite sets are closed. (Contributed by Mario Carneiro, 25-Dec-2016.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ⊆ 𝑋 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ (Clsd‘𝐽)) | ||
Theorem | hausnei 23293* | Neighborhood property of a Hausdorff space. (Contributed by NM, 8-Mar-2007.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Haus ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋 ∧ 𝑃 ≠ 𝑄)) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) | ||
Theorem | t0top 23294 | A T0 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.) |
⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ Top) | ||
Theorem | t1top 23295 | A T1 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.) |
⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top) | ||
Theorem | haustop 23296 | A Hausdorff space is a topology. (Contributed by NM, 5-Mar-2007.) |
⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top) | ||
Theorem | isreg 23297* | 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 ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))) | ||
Theorem | regtop 23298 | A regular space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.) |
⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top) | ||
Theorem | regsep 23299* | In a regular space, every neighborhood of a point contains a closed subneighborhood. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ ((𝐽 ∈ Reg ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈) → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)) | ||
Theorem | isnrm 23300* | 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 ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝐽 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))) |
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