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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | cnpdis 23301 | 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 23302 | 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 23303 | 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 23304 | 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 23305 | 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 23306 | 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 23307 | 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 23308 | 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 23309* | 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 23310* | 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 23311* | 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 23312 | 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 23313 | 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 23314 | Extend class notation with the class of all T0 spaces. |
| class Kol2 | ||
| Syntax | ct1 23315 | Extend class notation to include T1 spaces (also called Fréchet spaces). |
| class Fre | ||
| Syntax | cha 23316 | Extend class notation with the class of all Hausdorff spaces. |
| class Haus | ||
| Syntax | creg 23317 | Extend class notation with the class of all regular topologies. |
| class Reg | ||
| Syntax | cnrm 23318 | Extend class notation with the class of all normal topologies. |
| class Nrm | ||
| Syntax | ccnrm 23319 | Extend class notation with the class of all completely normal topologies. |
| class CNrm | ||
| Syntax | cpnrm 23320 | Extend class notation with the class of all perfectly normal topologies. |
| class PNrm | ||
| Definition | df-t0 23321* | Define T0 or Kolmogorov spaces. A T0 space satisfies a kind of "topological extensionality" principle (compare ax-ext 2708): 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 23355) 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 23322* | 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 23323* | 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 23324* | 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 23325* | 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 23326* | 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 23327* | 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 23328* | 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 23353. (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 23329* | 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 23330* | 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 23331* | 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 23332* | 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 23333* | 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 23334 | 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 23335 | 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 23336* | 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 23337 | A T0 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.) |
| ⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ Top) | ||
| Theorem | t1top 23338 | A T1 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.) |
| ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top) | ||
| Theorem | haustop 23339 | A Hausdorff space is a topology. (Contributed by NM, 5-Mar-2007.) |
| ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top) | ||
| Theorem | isreg 23340* | 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 23341 | A regular space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.) |
| ⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top) | ||
| Theorem | regsep 23342* | 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 23343* | 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))) | ||
| Theorem | nrmtop 23344 | A normal space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.) |
| ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top) | ||
| Theorem | cnrmtop 23345 | A completely normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.) |
| ⊢ (𝐽 ∈ CNrm → 𝐽 ∈ Top) | ||
| Theorem | iscnrm2 23346* | The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.) |
| ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 𝑋(𝐽 ↾t 𝑥) ∈ Nrm)) | ||
| Theorem | ispnrm 23347* | The property of being perfectly normal. (Contributed by Mario Carneiro, 26-Aug-2015.) |
| ⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓))) | ||
| Theorem | pnrmnrm 23348 | A perfectly normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.) |
| ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Nrm) | ||
| Theorem | pnrmtop 23349 | A perfectly normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.) |
| ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Top) | ||
| Theorem | pnrmcld 23350* | 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 ∈ (𝐽 ↑m ℕ)𝐴 = ∩ ran 𝑓) | ||
| Theorem | pnrmopn 23351* | An open set in a perfectly normal space is a countable union of closed sets. (Contributed by Mario Carneiro, 26-Aug-2015.) |
| ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)𝐴 = ∪ ran 𝑓) | ||
| Theorem | ist0-2 23352* | The predicate "is a T0 space". (Contributed by Mario Carneiro, 24-Aug-2015.) |
| ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))) | ||
| Theorem | ist0-3 23353* | The predicate "is a T0 space" expressed in more familiar terms. (Contributed by Jeff Hankins, 1-Feb-2010.) |
| ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 ((𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ∨ (¬ 𝑥 ∈ 𝑜 ∧ 𝑦 ∈ 𝑜))))) | ||
| Theorem | cnt0 23354 | The preimage of a T0 topology under an injective map is T0. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ ((𝐾 ∈ Kol2 ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Kol2) | ||
| Theorem | ist1-2 23355* | 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 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))) | ||
| Theorem | t1t0 23356 | A T1 space is a T0 space. (Contributed by Jeff Hankins, 1-Feb-2010.) |
| ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Kol2) | ||
| Theorem | ist1-3 23357* | 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.) |
| ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ 𝑋 ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥})) | ||
| Theorem | cnt1 23358 | The preimage of a T1 topology under an injective map is T1. (Contributed by Mario Carneiro, 26-Aug-2015.) |
| ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Fre) | ||
| Theorem | ishaus2 23359* | Express the predicate "𝐽 is a Hausdorff space." (Contributed by NM, 8-Mar-2007.) |
| ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Haus ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))) | ||
| Theorem | haust1 23360 | A Hausdorff space is a T1 space. (Contributed by FL, 11-Jun-2007.) (Proof shortened by Mario Carneiro, 24-Aug-2015.) |
| ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Fre) | ||
| Theorem | hausnei2 23361* | The Hausdorff condition still holds if one considers general neighborhoods instead of open sets. (Contributed by Jeff Hankins, 5-Sep-2009.) |
| ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Haus ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅))) | ||
| Theorem | cnhaus 23362 | The preimage of a Hausdorff topology under an injective map is Hausdorff. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ ((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Haus) | ||
| Theorem | nrmsep3 23363* | In a normal space, given a closed set 𝐵 inside an open set 𝐴, there is an open set 𝑥 such that 𝐵 ⊆ 𝑥 ⊆ cls(𝑥) ⊆ 𝐴. (Contributed by Mario Carneiro, 24-Aug-2015.) |
| ⊢ ((𝐽 ∈ Nrm ∧ (𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴)) → ∃𝑥 ∈ 𝐽 (𝐵 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)) | ||
| Theorem | nrmsep2 23364* | 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.) |
| ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅)) | ||
| Theorem | nrmsep 23365* | In a normal space, disjoint closed sets are separated by open sets. (Contributed by Jeff Hankins, 1-Feb-2010.) |
| ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) | ||
| Theorem | isnrm2 23366* | 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.) |
| ⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)))) | ||
| Theorem | isnrm3 23367* | A topological space is normal iff any two disjoint closed sets are separated by open sets. (Contributed by Mario Carneiro, 24-Aug-2015.) |
| ⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))) | ||
| Theorem | cnrmi 23368 | A subspace of a completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.) |
| ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Nrm) | ||
| Theorem | cnrmnrm 23369 | A completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.) |
| ⊢ (𝐽 ∈ CNrm → 𝐽 ∈ Nrm) | ||
| Theorem | restcnrm 23370 | A subspace of a completely normal space is completely normal. (Contributed by Mario Carneiro, 26-Aug-2015.) |
| ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ CNrm) | ||
| Theorem | resthauslem 23371 | Lemma for resthaus 23376 and similar theorems. If the topological property 𝐴 is preserved under injective preimages, then property 𝐴 passes to subspaces. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top) & ⊢ ((𝐽 ∈ 𝐴 ∧ ( I ↾ (𝑆 ∩ ∪ 𝐽)):(𝑆 ∩ ∪ 𝐽)–1-1→(𝑆 ∩ ∪ 𝐽) ∧ ( I ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t 𝑆) Cn 𝐽)) → (𝐽 ↾t 𝑆) ∈ 𝐴) ⇒ ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → (𝐽 ↾t 𝑆) ∈ 𝐴) | ||
| Theorem | lpcls 23372 | 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.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆)) | ||
| Theorem | perfcls 23373 | 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.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Perf ↔ (𝐽 ↾t ((cls‘𝐽)‘𝑆)) ∈ Perf)) | ||
| Theorem | restt0 23374 | A subspace of a T0 topology is T0. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ ((𝐽 ∈ Kol2 ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Kol2) | ||
| Theorem | restt1 23375 | A subspace of a T1 topology is T1. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Fre) | ||
| Theorem | resthaus 23376 | A subspace of a Hausdorff topology is Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.) (Proof shortened by Mario Carneiro, 25-Aug-2015.) |
| ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Haus) | ||
| Theorem | t1sep2 23377* | Any two points in a T1 space which have no separation are equal. (Contributed by Jeff Hankins, 1-Feb-2010.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → 𝐴 = 𝐵)) | ||
| Theorem | t1sep 23378* | Any two distinct points in a T1 space are separated by an open set. (Contributed by Jeff Hankins, 1-Feb-2010.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Fre ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ≠ 𝐵)) → ∃𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ∧ ¬ 𝐵 ∈ 𝑜)) | ||
| Theorem | sncld 23379 | A singleton is closed in a Hausdorff space. (Contributed by NM, 5-Mar-2007.) (Revised by Mario Carneiro, 24-Aug-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Haus ∧ 𝑃 ∈ 𝑋) → {𝑃} ∈ (Clsd‘𝐽)) | ||
| Theorem | sshauslem 23380 | Lemma for sshaus 23383 and similar theorems. If the topological property 𝐴 is preserved under injective preimages, then a topology finer than one with property 𝐴 also has property 𝐴. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top) & ⊢ ((𝐽 ∈ 𝐴 ∧ ( I ↾ 𝑋):𝑋–1-1→𝑋 ∧ ( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽)) → 𝐾 ∈ 𝐴) ⇒ ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ 𝐴) | ||
| Theorem | sst0 23381 | A topology finer than a T0 topology is T0. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Kol2 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ Kol2) | ||
| Theorem | sst1 23382 | A topology finer than a T1 topology is T1. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Fre ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ Fre) | ||
| Theorem | sshaus 23383 | A topology finer than a Hausdorff topology is Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Haus ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ Haus) | ||
| Theorem | regsep2 23384* | 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.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐴 ∈ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) | ||
| Theorem | isreg2 23385* | 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.) |
| ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Reg ↔ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥 ∈ 𝑋 (¬ 𝑥 ∈ 𝑐 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅)))) | ||
| Theorem | dnsconst 23386 | If a continuous mapping to a T1 space is constant on a dense subset, it is constant on the entire space. Note that ((cls‘𝐽)‘𝐴) = 𝑋 means "𝐴 is dense in 𝑋 " and 𝐴 ⊆ (◡𝐹 “ {𝑃}) means "𝐹 is constant on 𝐴 " (see funconstss 7076). (Contributed by NM, 15-Mar-2007.) (Proof shortened by Mario Carneiro, 21-Aug-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐹:𝑋⟶{𝑃}) | ||
| Theorem | ordtt1 23387 | The order topology is T1 for any poset. (Contributed by Mario Carneiro, 3-Sep-2015.) |
| ⊢ (𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ Fre) | ||
| Theorem | lmmo 23388 | 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.) |
| ⊢ (𝜑 → 𝐽 ∈ Haus) & ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝐴) & ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝐵) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
| Theorem | lmfun 23389 | The convergence relation is function-like in a Hausdorff space. (Contributed by Mario Carneiro, 26-Dec-2013.) |
| ⊢ (𝐽 ∈ Haus → Fun (⇝𝑡‘𝐽)) | ||
| Theorem | dishaus 23390 | 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.) |
| ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Haus) | ||
| Theorem | ordthauslem 23391* | Lemma for ordthaus 23392. (Contributed by Mario Carneiro, 13-Sep-2015.) |
| ⊢ 𝑋 = dom 𝑅 ⇒ ⊢ ((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 → (𝐴 ≠ 𝐵 → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴 ∈ 𝑚 ∧ 𝐵 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))) | ||
| Theorem | ordthaus 23392 | The order topology of a total order is Hausdorff. (Contributed by Mario Carneiro, 13-Sep-2015.) |
| ⊢ (𝑅 ∈ TosetRel → (ordTop‘𝑅) ∈ Haus) | ||
| Theorem | xrhaus 23393 | The topology of the extended reals is Hausdorff. (Contributed by Thierry Arnoux, 24-Mar-2017.) |
| ⊢ (ordTop‘ ≤ ) ∈ Haus | ||
| Syntax | ccmp 23394 | Extend class notation with the class of all compact spaces. |
| class Comp | ||
| Definition | df-cmp 23395* | Definition of a compact topology. A topology is compact iff any open covering of its underlying set contains a finite subcovering (Heine-Borel property). Definition C''' of [BourbakiTop1] p. I.59. Note: Bourbaki uses the term "quasi-compact" (saving "compact" for "compact Hausdorff"), but it is not the modern usage (which we follow). (Contributed by FL, 22-Dec-2008.) |
| ⊢ Comp = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)} | ||
| Theorem | iscmp 23396* | The predicate "is a compact topology". (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 11-Feb-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))) | ||
| Theorem | cmpcov 23397* | An open cover of a compact topology has a finite subcover. (Contributed by Jeff Hankins, 29-Jun-2009.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑆) → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = ∪ 𝑠) | ||
| Theorem | cmpcov2 23398* | Rewrite cmpcov 23397 for the cover {𝑦 ∈ 𝐽 ∣ 𝜑}. (Contributed by Mario Carneiro, 11-Sep-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Comp ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝜑)) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑠 ∧ ∀𝑦 ∈ 𝑠 𝜑)) | ||
| Theorem | cmpcovf 23399* | Combine cmpcov 23397 with ac6sfi 9320 to show the existence of a function that indexes the elements that are generating the open cover. (Contributed by Mario Carneiro, 14-Sep-2014.) |
| ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝑧 = (𝑓‘𝑦) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐽 ∈ Comp ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝐴 𝜑)) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶𝐴 ∧ ∀𝑦 ∈ 𝑠 𝜓))) | ||
| Theorem | cncmp 23400 | Compactness is respected by a continuous onto map. (Contributed by Jeff Hankins, 12-Jul-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.) |
| ⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ ((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Comp) | ||
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