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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | fincmp 23401 | A finite topology is compact. (Contributed by FL, 22-Dec-2008.) |
| ⊢ (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Comp) | ||
| Theorem | 0cmp 23402 | The singleton of the empty set is compact. (Contributed by FL, 2-Aug-2009.) |
| ⊢ {∅} ∈ Comp | ||
| Theorem | cmptop 23403 | A compact topology is a topology. (Contributed by Jeff Hankins, 29-Jun-2009.) |
| ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top) | ||
| Theorem | rncmp 23404 | 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 23405 | 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 23406 | 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 23407* | Lemma for cmpsub 23408. (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 23408* | 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 23409* | A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 24053, 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 23410 | 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 23411 | 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 23412* | 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 23413 | 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 23414* | Lemma for hauscmp 23415. (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 23415 | 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 23416* | 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 23417 | 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 23418* | 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 23419 | Extend class notation with the class of all connected topologies. |
| class Conn | ||
| Definition | df-conn 23420 | 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 23421 | 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 23422 | 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 23423 | 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 23424 | 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 23425 | 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 23426 | 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 23427* | 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 23428* | Connectedness for a subspace. See connsub 23429. (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 23429* | 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 23430 | 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 23431 | 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 23432 | 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 23433 | 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 23434 | 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 23435* | Lemma for iunconn 23436. (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 23436* | 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 23437 | 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 23438 | 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 23439* | 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 23440* | 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 23441* | 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 23442* | 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 23443* | 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 23444 | 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 23445 | Extend class definition to include the class of all first-countable topologies. |
| class 1stω | ||
| Syntax | c2ndc 23446 | Extend class definition to include the class of all second-countable topologies. |
| class 2ndω | ||
| Definition | df-1stc 23447* | 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 23448* | 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 23449* | 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 23450* | 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 23451 | A first-countable topology is a topology. (Contributed by Jeff Hankins, 22-Aug-2009.) |
| ⊢ (𝐽 ∈ 1stω → 𝐽 ∈ Top) | ||
| Theorem | 1stcclb 23452* | 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 23453* | 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 23454* | 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)) | ||
| Theorem | 2ndctop 23455 | A second-countable topology is a topology. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.) |
| ⊢ (𝐽 ∈ 2ndω → 𝐽 ∈ Top) | ||
| Theorem | 2ndci 23456 | A countable basis generates a second-countable topology. (Contributed by Mario Carneiro, 21-Mar-2015.) |
| ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → (topGen‘𝐵) ∈ 2ndω) | ||
| Theorem | 2ndcsb 23457* | Having a countable subbase is a sufficient condition for second-countability. (Contributed by Jeff Hankins, 17-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
| ⊢ (𝐽 ∈ 2ndω ↔ ∃𝑥(𝑥 ≼ ω ∧ (topGen‘(fi‘𝑥)) = 𝐽)) | ||
| Theorem | 2ndcredom 23458 | A second-countable space has at most the cardinality of the continuum. (Contributed by Mario Carneiro, 9-Apr-2015.) |
| ⊢ (𝐽 ∈ 2ndω → 𝐽 ≼ ℝ) | ||
| Theorem | 2ndc1stc 23459 | A second-countable space is first-countable. (Contributed by Jeff Hankins, 17-Jan-2010.) |
| ⊢ (𝐽 ∈ 2ndω → 𝐽 ∈ 1stω) | ||
| Theorem | 1stcrestlem 23460* | Lemma for 1stcrest 23461. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 30-Apr-2015.) |
| ⊢ (𝐵 ≼ ω → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) | ||
| Theorem | 1stcrest 23461 | A subspace of a first-countable space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.) |
| ⊢ ((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ 1stω) | ||
| Theorem | 2ndcrest 23462 | A subspace of a second-countable space is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.) |
| ⊢ ((𝐽 ∈ 2ndω ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ 2ndω) | ||
| Theorem | 2ndcctbss 23463* | If a topology is second-countable, every base has a countable subset which is a base. Exercise 16B2 in Willard. (Contributed by Jeff Hankins, 28-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
| ⊢ 𝐽 = (topGen‘𝐵) & ⊢ 𝑆 = {〈𝑢, 𝑣〉 ∣ (𝑢 ∈ 𝑐 ∧ 𝑣 ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 (𝑢 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑣))} ⇒ ⊢ ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏 ⊆ 𝐵 ∧ 𝐽 = (topGen‘𝑏))) | ||
| Theorem | 2ndcdisj 23464* | Any disjoint family of open sets in a second-countable space is countable. (The sets are required to be nonempty because otherwise there could be many empty sets in the family.) (Contributed by Mario Carneiro, 21-Mar-2015.) (Proof shortened by Mario Carneiro, 9-Apr-2015.) (Revised by NM, 17-Jun-2017.) |
| ⊢ ((𝐽 ∈ 2ndω ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → 𝐴 ≼ ω) | ||
| Theorem | 2ndcdisj2 23465* | Any disjoint collection of open sets in a second-countable space is countable. (Contributed by Mario Carneiro, 21-Mar-2015.) (Proof shortened by Mario Carneiro, 9-Apr-2015.) (Revised by NM, 17-Jun-2017.) |
| ⊢ ((𝐽 ∈ 2ndω ∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) → 𝐴 ≼ ω) | ||
| Theorem | 2ndcomap 23466* | A surjective continuous open map maps second-countable spaces to second-countable spaces. (Contributed by Mario Carneiro, 9-Apr-2015.) |
| ⊢ 𝑌 = ∪ 𝐾 & ⊢ (𝜑 → 𝐽 ∈ 2ndω) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → ran 𝐹 = 𝑌) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐾) ⇒ ⊢ (𝜑 → 𝐾 ∈ 2ndω) | ||
| Theorem | 2ndcsep 23467* | A second-countable topology is separable, which is to say it contains a countable dense subset. (Contributed by Mario Carneiro, 13-Apr-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ 2ndω → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)) | ||
| Theorem | dis2ndc 23468 | A discrete space is second-countable iff it is countable. (Contributed by Mario Carneiro, 13-Apr-2015.) |
| ⊢ (𝑋 ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω) | ||
| Theorem | 1stcelcls 23469* | A point belongs to the closure of a subset iff there is a sequence in the subset converging to it. Theorem 1.4-6(a) of [Kreyszig] p. 30. This proof uses countable choice ax-cc 10475. A space satisfying the conclusion of this theorem is called a sequential space, so the theorem can also be stated as "every first-countable space is a sequential space". (Contributed by Mario Carneiro, 21-Mar-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ 1stω ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃))) | ||
| Theorem | 1stccnp 23470* | A mapping is continuous at 𝑃 in a first-countable space 𝑋 iff it is sequentially continuous at 𝑃, meaning that the image under 𝐹 of every sequence converging at 𝑃 converges to 𝐹(𝑃). This proof uses ax-cc 10475, but only via 1stcelcls 23469, so it could be refactored into a proof that continuity and sequential continuity are the same in sequential spaces. (Contributed by Mario Carneiro, 7-Sep-2015.) |
| ⊢ (𝜑 → 𝐽 ∈ 1stω) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝑃 ∈ 𝑋) ⇒ ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋 ∧ 𝑓(⇝𝑡‘𝐽)𝑃) → (𝐹 ∘ 𝑓)(⇝𝑡‘𝐾)(𝐹‘𝑃))))) | ||
| Theorem | 1stccn 23471* | A mapping 𝑋⟶𝑌, where 𝑋 is first-countable, is continuous iff it is sequentially continuous, meaning that for any sequence 𝑓(𝑛) converging to 𝑥, its image under 𝐹 converges to 𝐹(𝑥). (Contributed by Mario Carneiro, 7-Sep-2015.) |
| ⊢ (𝜑 → 𝐽 ∈ 1stω) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑓(𝑓:ℕ⟶𝑋 → ∀𝑥(𝑓(⇝𝑡‘𝐽)𝑥 → (𝐹 ∘ 𝑓)(⇝𝑡‘𝐾)(𝐹‘𝑥))))) | ||
| Syntax | clly 23472 | Extend class notation with the "locally 𝐴 " predicate of a topological space. |
| class Locally 𝐴 | ||
| Syntax | cnlly 23473 | Extend class notation with the "N-locally 𝐴 " predicate of a topological space. |
| class 𝑛-Locally 𝐴 | ||
| Definition | df-lly 23474* | Define a space that is locally 𝐴, where 𝐴 is a topological property like "compact", "connected", or "path-connected". A topological space is locally 𝐴 if every neighborhood of a point contains an open subneighborhood that is 𝐴 in the subspace topology. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴)} | ||
| Definition | df-nlly 23475* |
Define a space that is n-locally 𝐴, where 𝐴 is a topological
property like "compact", "connected", or
"path-connected". A
topological space is n-locally 𝐴 if every neighborhood of a point
contains a subneighborhood that is 𝐴 in the subspace topology.
The terminology "n-locally", where 'n' stands for "neighborhood", is not standard, although this is sometimes called "weakly locally 𝐴". The reason for the distinction is that some notions only make sense for arbitrary neighborhoods (such as "locally compact", which is actually 𝑛-Locally Comp in our terminology - open compact sets are not very useful), while others such as "locally connected" are strictly weaker notions if the neighborhoods are not required to be open. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ 𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴} | ||
| Theorem | islly 23476* | The property of being a locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) | ||
| Theorem | isnlly 23477* | The property of being an n-locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴)) | ||
| Theorem | llyeq 23478 | Equality theorem for the Locally 𝐴 predicate. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ (𝐴 = 𝐵 → Locally 𝐴 = Locally 𝐵) | ||
| Theorem | nllyeq 23479 | Equality theorem for the Locally 𝐴 predicate. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ (𝐴 = 𝐵 → 𝑛-Locally 𝐴 = 𝑛-Locally 𝐵) | ||
| Theorem | llytop 23480 | A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ Top) | ||
| Theorem | nllytop 23481 | A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top) | ||
| Theorem | llyi 23482* | The property of a locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑢 ∈ 𝐽 (𝑢 ⊆ 𝑈 ∧ 𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)) | ||
| Theorem | nllyi 23483* | The property of an n-locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑃})(𝑢 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)) | ||
| Theorem | nlly2i 23484* | Eliminate the neighborhood symbol from nllyi 23483. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑠 ∈ 𝒫 𝑈∃𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴)) | ||
| Theorem | llynlly 23485 | A locally 𝐴 space is n-locally 𝐴: the "n-locally" predicate is the weaker notion. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ 𝑛-Locally 𝐴) | ||
| Theorem | llyssnlly 23486 | A locally 𝐴 space is n-locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ Locally 𝐴 ⊆ 𝑛-Locally 𝐴 | ||
| Theorem | llyss 23487 | The "locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ (𝐴 ⊆ 𝐵 → Locally 𝐴 ⊆ Locally 𝐵) | ||
| Theorem | nllyss 23488 | The "n-locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ (𝐴 ⊆ 𝐵 → 𝑛-Locally 𝐴 ⊆ 𝑛-Locally 𝐵) | ||
| Theorem | subislly 23489* | The property of a subspace being locally 𝐴. (Contributed by Mario Carneiro, 10-Mar-2015.) |
| ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → ((𝐽 ↾t 𝐵) ∈ Locally 𝐴 ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ (𝑥 ∩ 𝐵)∃𝑢 ∈ 𝐽 ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t (𝑢 ∩ 𝐵)) ∈ 𝐴))) | ||
| Theorem | restnlly 23490* | If the property 𝐴 passes to open subspaces, then a space is n-locally 𝐴 iff it is locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝑛-Locally 𝐴 = Locally 𝐴) | ||
| Theorem | restlly 23491* | If the property 𝐴 passes to open subspaces, then a space which is 𝐴 is also locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴) & ⊢ (𝜑 → 𝐴 ⊆ Top) ⇒ ⊢ (𝜑 → 𝐴 ⊆ Locally 𝐴) | ||
| Theorem | islly2 23492* | An alternative expression for 𝐽 ∈ Locally 𝐴 when 𝐴 passes to open subspaces: A space is locally 𝐴 if every point is contained in an open neighborhood with property 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴) & ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝜑 → (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))) | ||
| Theorem | llyrest 23493 | An open subspace of a locally 𝐴 space is also locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ Locally 𝐴) | ||
| Theorem | nllyrest 23494 | An open subspace of an n-locally 𝐴 space is also n-locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ 𝑛-Locally 𝐴) | ||
| Theorem | loclly 23495 | If 𝐴 is a local property, then both Locally 𝐴 and 𝑛-Locally 𝐴 simplify to 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ (Locally 𝐴 = 𝐴 ↔ 𝑛-Locally 𝐴 = 𝐴) | ||
| Theorem | llyidm 23496 | Idempotence of the "locally" predicate, i.e. being "locally 𝐴 " is a local property. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ Locally Locally 𝐴 = Locally 𝐴 | ||
| Theorem | nllyidm 23497 | Idempotence of the "n-locally" predicate, i.e. being "n-locally 𝐴 " is a local property. (Use loclly 23495 to show 𝑛-Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴.) (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴 | ||
| Theorem | toplly 23498 | A topology is locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ Locally Top = Top | ||
| Theorem | topnlly 23499 | A topology is n-locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ 𝑛-Locally Top = Top | ||
| Theorem | hauslly 23500 | A Hausdorff space is locally Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Locally Haus) | ||
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