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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ist1-2 22101* | 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 22102 | A T1 space is a T0 space. (Contributed by Jeff Hankins, 1-Feb-2010.) |
⊢ (𝐽 ∈ Fre → 𝐽 ∈ Kol2) | ||
Theorem | ist1-3 22103* | 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 22104 | 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 22105* | 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 22106 | 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 22107* | 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 22108 | 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 22109* | 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 22110* | 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 22111* | 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 22112* | 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 22113* | 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 22114 | 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 22115 | A completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.) |
⊢ (𝐽 ∈ CNrm → 𝐽 ∈ Nrm) | ||
Theorem | restcnrm 22116 | 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 22117 | Lemma for resthaus 22122 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 22118 | 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 22119 | 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 22120 | 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 22121 | 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 22122 | 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 22123* | 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 22124* | 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 22125 | 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 22126 | Lemma for sshaus 22129 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 22127 | 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 22128 | 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 22129 | 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 22130* | 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 22131* | 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 22132 | 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 6836). (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 22133 | The order topology is T1 for any poset. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ Fre) | ||
Theorem | lmmo 22134 | 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 22135 | The convergence relation is function-like in a Hausdorff space. (Contributed by Mario Carneiro, 26-Dec-2013.) |
⊢ (𝐽 ∈ Haus → Fun (⇝𝑡‘𝐽)) | ||
Theorem | dishaus 22136 | 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 22137* | Lemma for ordthaus 22138. (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 22138 | The order topology of a total order is Hausdorff. (Contributed by Mario Carneiro, 13-Sep-2015.) |
⊢ (𝑅 ∈ TosetRel → (ordTop‘𝑅) ∈ Haus) | ||
Theorem | xrhaus 22139 | The topology of the extended reals is Hausdorff. (Contributed by Thierry Arnoux, 24-Mar-2017.) |
⊢ (ordTop‘ ≤ ) ∈ Haus | ||
Syntax | ccmp 22140 | Extend class notation with the class of all compact spaces. |
class Comp | ||
Definition | df-cmp 22141* | 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 22142* | 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 22143* | 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 22144* | Rewrite cmpcov 22143 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 22145* | Combine cmpcov 22143 with ac6sfi 8839 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 22146 | 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) | ||
Theorem | fincmp 22147 | A finite topology is compact. (Contributed by FL, 22-Dec-2008.) |
⊢ (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Comp) | ||
Theorem | 0cmp 22148 | The singleton of the empty set is compact. (Contributed by FL, 2-Aug-2009.) |
⊢ {∅} ∈ Comp | ||
Theorem | cmptop 22149 | A compact topology is a topology. (Contributed by Jeff Hankins, 29-Jun-2009.) |
⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top) | ||
Theorem | rncmp 22150 | The image of a compact set under a continuous function is compact. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐾 ↾t ran 𝐹) ∈ Comp) | ||
Theorem | imacmp 22151 | The image of a compact set under a continuous function is compact. (Contributed by Mario Carneiro, 18-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.) |
⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐾 ↾t (𝐹 “ 𝐴)) ∈ Comp) | ||
Theorem | discmp 22152 | A discrete topology is compact iff the base set is finite. (Contributed by Mario Carneiro, 19-Mar-2015.) |
⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Comp) | ||
Theorem | cmpsublem 22153* | Lemma for cmpsub 22154. (Contributed by Jeff Hankins, 28-Jun-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∀𝑐 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑) → ∀𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)(∪ (𝐽 ↾t 𝑆) = ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡))) | ||
Theorem | cmpsub 22154* | Two equivalent ways of describing a compact subset of a topological space. Inspired by Sue E. Goodman's Beginning Topology. (Contributed by Jeff Hankins, 22-Jun-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Comp ↔ ∀𝑐 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑))) | ||
Theorem | tgcmp 22155* | A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 22799, which further specializes to subbases, assuming the ultrafilter lemma.) (Contributed by Mario Carneiro, 26-Aug-2015.) |
⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))) | ||
Theorem | cmpcld 22156 | A closed subset of a compact space is compact. (Contributed by Jeff Hankins, 29-Jun-2009.) |
⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝐽 ↾t 𝑆) ∈ Comp) | ||
Theorem | uncmp 22157 | The union of two compact sets is compact. (Contributed by Jeff Hankins, 30-Jan-2010.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ ((𝐽 ↾t 𝑆) ∈ Comp ∧ (𝐽 ↾t 𝑇) ∈ Comp)) → 𝐽 ∈ Comp) | ||
Theorem | fiuncmp 22158* | A finite union of compact sets is compact. (Contributed by Mario Carneiro, 19-Mar-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾t 𝐵) ∈ Comp) → (𝐽 ↾t ∪ 𝑥 ∈ 𝐴 𝐵) ∈ Comp) | ||
Theorem | sscmp 22159 | A subset of a compact topology (i.e. a coarser topology) is compact. (Contributed by Mario Carneiro, 20-Mar-2015.) |
⊢ 𝑋 = ∪ 𝐾 ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ Comp) | ||
Theorem | hauscmplem 22160* | Lemma for hauscmp 22161. (Contributed by Mario Carneiro, 27-Nov-2013.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑂 = {𝑦 ∈ 𝐽 ∣ ∃𝑤 ∈ 𝐽 (𝐴 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋 ∖ 𝑦))} & ⊢ (𝜑 → 𝐽 ∈ Haus) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) & ⊢ (𝜑 → (𝐽 ↾t 𝑆) ∈ Comp) & ⊢ (𝜑 → 𝐴 ∈ (𝑋 ∖ 𝑆)) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋 ∖ 𝑆))) | ||
Theorem | hauscmp 22161 | A compact subspace of a T2 space is closed. (Contributed by Jeff Hankins, 16-Jan-2010.) (Proof shortened by Mario Carneiro, 14-Dec-2013.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Haus ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → 𝑆 ∈ (Clsd‘𝐽)) | ||
Theorem | cmpfi 22162* | If a topology is compact and a collection of closed sets has the finite intersection property, its intersection is nonempty. (Contributed by Jeff Hankins, 25-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.) |
⊢ (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅))) | ||
Theorem | cmpfii 22163 | In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ ((𝐽 ∈ Comp ∧ 𝑋 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑋)) → ∩ 𝑋 ≠ ∅) | ||
Theorem | bwth 22164* | The glorious Bolzano-Weierstrass theorem. The first general topology theorem ever proved. The first mention of this theorem can be found in a course by Weierstrass from 1865. In his course Weierstrass called it a lemma. He didn't know how famous this theorem would be. He used a Euclidean space instead of a general compact space. And he was not aware of the Heine-Borel property. But the concepts of neighborhood and limit point were already there although not precisely defined. Cantor was one of his students. He published and used the theorem in an article from 1872. The rest of the general topology followed from that. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) Revised by BL to significantly shorten the proof and avoid infinity, regularity, and choice. (Revised by Brendan Leahy, 26-Dec-2018.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Comp ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → ∃𝑥 ∈ 𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴)) | ||
Syntax | cconn 22165 | Extend class notation with the class of all connected topologies. |
class Conn | ||
Definition | df-conn 22166 | Topologies are connected when only ∅ and ∪ 𝑗 are both open and closed. (Contributed by FL, 17-Nov-2008.) |
⊢ Conn = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, ∪ 𝑗}} | ||
Theorem | isconn 22167 | The predicate 𝐽 is a connected topology . (Contributed by FL, 17-Nov-2008.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})) | ||
Theorem | isconn2 22168 | The predicate 𝐽 is a connected topology . (Contributed by Mario Carneiro, 10-Mar-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋})) | ||
Theorem | connclo 22169 | The only nonempty clopen set of a connected topology is the whole space. (Contributed by Mario Carneiro, 10-Mar-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐽 ∈ Conn) & ⊢ (𝜑 → 𝐴 ∈ 𝐽) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽)) ⇒ ⊢ (𝜑 → 𝐴 = 𝑋) | ||
Theorem | conndisj 22170 | If a topology is connected, its underlying set can't be partitioned into two nonempty non-overlapping open sets. (Contributed by FL, 16-Nov-2008.) (Proof shortened by Mario Carneiro, 10-Mar-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐽 ∈ Conn) & ⊢ (𝜑 → 𝐴 ∈ 𝐽) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → 𝐵 ∈ 𝐽) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) ⇒ ⊢ (𝜑 → (𝐴 ∪ 𝐵) ≠ 𝑋) | ||
Theorem | conntop 22171 | A connected topology is a topology. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 14-Dec-2013.) |
⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top) | ||
Theorem | indisconn 22172 | The indiscrete topology (or trivial topology) on any set is connected. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ {∅, 𝐴} ∈ Conn | ||
Theorem | dfconn2 22173* | An alternate definition of connectedness. (Contributed by Jeff Hankins, 9-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Conn ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ 𝑋))) | ||
Theorem | connsuba 22174* | Connectedness for a subspace. See connsub 22175. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 10-Mar-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((𝐽 ↾t 𝐴) ∈ Conn ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (((𝑥 ∩ 𝐴) ≠ ∅ ∧ (𝑦 ∩ 𝐴) ≠ ∅ ∧ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅) → ((𝑥 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴))) | ||
Theorem | connsub 22175* | Two equivalent ways of saying that a subspace topology is connected. (Contributed by Jeff Hankins, 9-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Conn ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (((𝑥 ∩ 𝑆) ≠ ∅ ∧ (𝑦 ∩ 𝑆) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ 𝑆)) → ¬ 𝑆 ⊆ (𝑥 ∪ 𝑦)))) | ||
Theorem | cnconn 22176 | Connectedness is respected by a continuous onto map. (Contributed by Jeff Hankins, 12-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.) |
⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ ((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Conn) | ||
Theorem | nconnsubb 22177 | Disconnectedness for a subspace. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 10-Mar-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → 𝑈 ∈ 𝐽) & ⊢ (𝜑 → 𝑉 ∈ 𝐽) & ⊢ (𝜑 → (𝑈 ∩ 𝐴) ≠ ∅) & ⊢ (𝜑 → (𝑉 ∩ 𝐴) ≠ ∅) & ⊢ (𝜑 → ((𝑈 ∩ 𝑉) ∩ 𝐴) = ∅) & ⊢ (𝜑 → 𝐴 ⊆ (𝑈 ∪ 𝑉)) ⇒ ⊢ (𝜑 → ¬ (𝐽 ↾t 𝐴) ∈ Conn) | ||
Theorem | connsubclo 22178 | If a clopen set meets a connected subspace, it must contain the entire subspace. (Contributed by Mario Carneiro, 10-Mar-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → (𝐽 ↾t 𝐴) ∈ Conn) & ⊢ (𝜑 → 𝐵 ∈ 𝐽) & ⊢ (𝜑 → (𝐵 ∩ 𝐴) ≠ ∅) & ⊢ (𝜑 → 𝐵 ∈ (Clsd‘𝐽)) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | ||
Theorem | connima 22179 | The image of a connected set is connected. (Contributed by Mario Carneiro, 7-Jul-2015.) (Revised by Mario Carneiro, 22-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → (𝐽 ↾t 𝐴) ∈ Conn) ⇒ ⊢ (𝜑 → (𝐾 ↾t (𝐹 “ 𝐴)) ∈ Conn) | ||
Theorem | conncn 22180 | A continuous function from a connected topology with one point in a clopen set must lie entirely within the set. (Contributed by Mario Carneiro, 16-Feb-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐽 ∈ Conn) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝑈 ∈ 𝐾) & ⊢ (𝜑 → 𝑈 ∈ (Clsd‘𝐾)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝑈) ⇒ ⊢ (𝜑 → 𝐹:𝑋⟶𝑈) | ||
Theorem | iunconnlem 22181* | Lemma for iunconn 22182. (Contributed by Mario Carneiro, 11-Jun-2014.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ⊆ 𝑋) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑃 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐽 ↾t 𝐵) ∈ Conn) & ⊢ (𝜑 → 𝑈 ∈ 𝐽) & ⊢ (𝜑 → 𝑉 ∈ 𝐽) & ⊢ (𝜑 → (𝑉 ∩ ∪ 𝑘 ∈ 𝐴 𝐵) ≠ ∅) & ⊢ (𝜑 → (𝑈 ∩ 𝑉) ⊆ (𝑋 ∖ ∪ 𝑘 ∈ 𝐴 𝐵)) & ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝐵 ⊆ (𝑈 ∪ 𝑉)) & ⊢ Ⅎ𝑘𝜑 ⇒ ⊢ (𝜑 → ¬ 𝑃 ∈ 𝑈) | ||
Theorem | iunconn 22182* | The indexed union of connected overlapping subspaces sharing a common point is connected. (Contributed by Mario Carneiro, 11-Jun-2014.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ⊆ 𝑋) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑃 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐽 ↾t 𝐵) ∈ Conn) ⇒ ⊢ (𝜑 → (𝐽 ↾t ∪ 𝑘 ∈ 𝐴 𝐵) ∈ Conn) | ||
Theorem | unconn 22183 | The union of two connected overlapping subspaces is connected. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 11-Jun-2014.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐽 ↾t 𝐵) ∈ Conn) → (𝐽 ↾t (𝐴 ∪ 𝐵)) ∈ Conn)) | ||
Theorem | clsconn 22184 | The closure of a connected set is connected. (Contributed by Mario Carneiro, 19-Mar-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) → (𝐽 ↾t ((cls‘𝐽)‘𝐴)) ∈ Conn) | ||
Theorem | conncompid 22185* | The connected component containing 𝐴 contains 𝐴. (Contributed by Mario Carneiro, 19-Mar-2015.) |
⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑆) | ||
Theorem | conncompconn 22186* | The connected component containing 𝐴 is connected. (Contributed by Mario Carneiro, 19-Mar-2015.) |
⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐽 ↾t 𝑆) ∈ Conn) | ||
Theorem | conncompss 22187* | The connected component containing 𝐴 is a superset of any other connected set containing 𝐴. (Contributed by Mario Carneiro, 19-Mar-2015.) |
⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} ⇒ ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ⊆ 𝑆) | ||
Theorem | conncompcld 22188* | The connected component containing 𝐴 is a closed set. (Contributed by Mario Carneiro, 19-Mar-2015.) |
⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝑆 ∈ (Clsd‘𝐽)) | ||
Theorem | conncompclo 22189* | The connected component containing 𝐴 is a subset of any clopen set containing 𝐴. (Contributed by Mario Carneiro, 20-Sep-2015.) |
⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑆 ⊆ 𝑇) | ||
Theorem | t1connperf 22190 | A connected T1 space is perfect, unless it is the topology of a singleton. (Contributed by Mario Carneiro, 26-Dec-2016.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn ∧ ¬ 𝑋 ≈ 1o) → 𝐽 ∈ Perf) | ||
Syntax | c1stc 22191 | Extend class definition to include the class of all first-countable topologies. |
class 1stω | ||
Syntax | c2ndc 22192 | Extend class definition to include the class of all second-countable topologies. |
class 2ndω | ||
Definition | df-1stc 22193* | Define the class of all first-countable topologies. (Contributed by Jeff Hankins, 22-Aug-2009.) |
⊢ 1stω = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧)))} | ||
Definition | df-2ndc 22194* | Define the class of all second-countable topologies. (Contributed by Jeff Hankins, 17-Jan-2010.) |
⊢ 2ndω = {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)} | ||
Theorem | is1stc 22195* | The predicate "is a first-countable topology." This can be described as "every point has a countable local basis" - that is, every point has a countable collection of open sets containing it such that every open set containing the point has an open set from this collection as a subset. (Contributed by Jeff Hankins, 22-Aug-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ 1stω ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧))))) | ||
Theorem | is1stc2 22196* | An equivalent way of saying "is a first-countable topology." (Contributed by Jeff Hankins, 22-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ 1stω ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) | ||
Theorem | 1stctop 22197 | A first-countable topology is a topology. (Contributed by Jeff Hankins, 22-Aug-2009.) |
⊢ (𝐽 ∈ 1stω → 𝐽 ∈ Top) | ||
Theorem | 1stcclb 22198* | A property of points in a first-countable topology. (Contributed by Jeff Hankins, 22-Aug-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))) | ||
Theorem | 1stcfb 22199* | For any point 𝐴 in a first-countable topology, there is a function 𝑓:ℕ⟶𝐽 enumerating neighborhoods of 𝐴 which is decreasing and forms a local base. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓‘𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘)) ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦))) | ||
Theorem | is2ndc 22200* | The property of being second-countable. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.) |
⊢ (𝐽 ∈ 2ndω ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) |
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