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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cmpcov2 22001* | Rewrite cmpcov 22000 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 22002* | Combine cmpcov 22000 with ac6sfi 8765 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 22003 | 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 22004 | A finite topology is compact. (Contributed by FL, 22-Dec-2008.) |
⊢ (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Comp) | ||
Theorem | 0cmp 22005 | The singleton of the empty set is compact. (Contributed by FL, 2-Aug-2009.) |
⊢ {∅} ∈ Comp | ||
Theorem | cmptop 22006 | A compact topology is a topology. (Contributed by Jeff Hankins, 29-Jun-2009.) |
⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top) | ||
Theorem | rncmp 22007 | 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 22008 | 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 22009 | 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 22010* | Lemma for cmpsub 22011. (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 22011* | 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 22012* | A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 22656, 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 22013 | 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 22014 | 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 22015* | 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 22016 | 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 22017* | Lemma for hauscmp 22018. (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 22018 | 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 22019* | 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 22020 | 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 22021* | 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 22022 | Extend class notation with the class of all connected topologies. |
class Conn | ||
Definition | df-conn 22023 | 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 22024 | 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 22025 | 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 22026 | 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 22027 | 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 22028 | 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 22029 | 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 22030* | 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 22031* | Connectedness for a subspace. See connsub 22032. (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 22032* | 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 22033 | 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 22034 | 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 22035 | 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 22036 | 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 22037 | 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 22038* | Lemma for iunconn 22039. (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 22039* | 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 22040 | 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 22041 | 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 22042* | 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 22043* | 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 22044* | 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 22045* | 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 22046* | 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 22047 | 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 22048 | Extend class definition to include the class of all first-countable topologies. |
class 1stω | ||
Syntax | c2ndc 22049 | Extend class definition to include the class of all second-countable topologies. |
class 2ndω | ||
Definition | df-1stc 22050* | 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 22051* | 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 22052* | 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 22053* | 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 22054 | A first-countable topology is a topology. (Contributed by Jeff Hankins, 22-Aug-2009.) |
⊢ (𝐽 ∈ 1stω → 𝐽 ∈ Top) | ||
Theorem | 1stcclb 22055* | 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 22056* | 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 22057* | 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 22058 | 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 22059 | A countable basis generates a second-countable topology. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → (topGen‘𝐵) ∈ 2ndω) | ||
Theorem | 2ndcsb 22060* | 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 22061 | 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 22062 | A second-countable space is first-countable. (Contributed by Jeff Hankins, 17-Jan-2010.) |
⊢ (𝐽 ∈ 2ndω → 𝐽 ∈ 1stω) | ||
Theorem | 1stcrestlem 22063* | Lemma for 1stcrest 22064. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ (𝐵 ≼ ω → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) | ||
Theorem | 1stcrest 22064 | 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 22065 | 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 22066* | 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 22067* | 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 22068* | 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 22069* | 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 22070* | 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 22071 | A discrete space is second-countable iff it is countable. (Contributed by Mario Carneiro, 13-Apr-2015.) |
⊢ (𝑋 ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω) | ||
Theorem | 1stcelcls 22072* | 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 9860. 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 22073* | 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 9860, but only via 1stcelcls 22072, 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 22074* | 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 22075 | Extend class notation with the "locally 𝐴 " predicate of a topological space. |
class Locally 𝐴 | ||
Syntax | cnlly 22076 | Extend class notation with the "N-locally 𝐴 " predicate of a topological space. |
class 𝑛-Locally 𝐴 | ||
Definition | df-lly 22077* | 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 22078* |
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 22079* | 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 22080* | 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 22081 | Equality theorem for the Locally 𝐴 predicate. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ (𝐴 = 𝐵 → Locally 𝐴 = Locally 𝐵) | ||
Theorem | nllyeq 22082 | Equality theorem for the Locally 𝐴 predicate. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ (𝐴 = 𝐵 → 𝑛-Locally 𝐴 = 𝑛-Locally 𝐵) | ||
Theorem | llytop 22083 | A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ Top) | ||
Theorem | nllytop 22084 | A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top) | ||
Theorem | llyi 22085* | 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 22086* | 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 22087* | Eliminate the neighborhood symbol from nllyi 22086. (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 22088 | 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 22089 | A locally 𝐴 space is n-locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ Locally 𝐴 ⊆ 𝑛-Locally 𝐴 | ||
Theorem | llyss 22090 | The "locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ (𝐴 ⊆ 𝐵 → Locally 𝐴 ⊆ Locally 𝐵) | ||
Theorem | nllyss 22091 | The "n-locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ (𝐴 ⊆ 𝐵 → 𝑛-Locally 𝐴 ⊆ 𝑛-Locally 𝐵) | ||
Theorem | subislly 22092* | 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 22093* | 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 22094* | 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 22095* | 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 22096 | 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 22097 | 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 22098 | 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 22099 | 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 22100 | Idempotence of the "n-locally" predicate, i.e. being "n-locally 𝐴 " is a local property. (Use loclly 22098 to show 𝑛-Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴.) (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴 |
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