P. 234 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23301-23400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	hauscmp 23301	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 23302*	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 23303	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 ≠ ∅)
12.1.12  Bolzano-Weierstrass theorem
Theorem	bwth 23304*	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))
12.1.13  Connectedness
Syntax	cconn 23305	Extend class notation with the class of all connected topologies.
class Conn
Definition	df-conn 23306	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 23307	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 23308	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 23309	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 23310	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 23311	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 23312	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 23313*	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 23314*	Connectedness for a subspace. See connsub 23315. (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 23315*	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 23316	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 23317	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 23318	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 23319	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 23320	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 23321*	Lemma for iunconn 23322. (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 23322*	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 23323	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 23324	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 23325*	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 23326*	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 23327*	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 23328*	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 23329*	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 23330	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)
12.1.14  First- and second-countability
Syntax	c1stc 23331	Extend class definition to include the class of all first-countable topologies.
class 1stω
Syntax	c2ndc 23332	Extend class definition to include the class of all second-countable topologies.
class 2ndω
Definition	df-1stc 23333*	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 23334*	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 23335*	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 23336*	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 23337	A first-countable topology is a topology. (Contributed by Jeff Hankins, 22-Aug-2009.)
⊢ (𝐽 ∈ 1stω → 𝐽 ∈ Top)
Theorem	1stcclb 23338*	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 23339*	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 23340*	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 23341	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 23342	A countable basis generates a second-countable topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → (topGen‘𝐵) ∈ 2ndω)
Theorem	2ndcsb 23343*	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 23344	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 23345	A second-countable space is first-countable. (Contributed by Jeff Hankins, 17-Jan-2010.)
⊢ (𝐽 ∈ 2ndω → 𝐽 ∈ 1stω)
Theorem	1stcrestlem 23346*	Lemma for 1stcrest 23347. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ (𝐵 ≼ ω → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω)
Theorem	1stcrest 23347	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 23348	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 23349*	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 23350*	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 23351*	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 23352*	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 23353*	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 23354	A discrete space is second-countable iff it is countable. (Contributed by Mario Carneiro, 13-Apr-2015.)
⊢ (𝑋 ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω)
Theorem	1stcelcls 23355*	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 10395. 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 23356*	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 10395, but only via 1stcelcls 23355, 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 23357*	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)))))
12.1.15  Local topological properties
Syntax	clly 23358	Extend class notation with the "locally 𝐴 " predicate of a topological space.
class Locally 𝐴
Syntax	cnlly 23359	Extend class notation with the "N-locally 𝐴 " predicate of a topological space.
class 𝑛-Locally 𝐴
Definition	df-lly 23360*	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 23361*	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 23362*	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 23363*	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 23364	Equality theorem for the Locally 𝐴 predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
⊢ (𝐴 = 𝐵 → Locally 𝐴 = Locally 𝐵)
Theorem	nllyeq 23365	Equality theorem for the Locally 𝐴 predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
⊢ (𝐴 = 𝐵 → 𝑛-Locally 𝐴 = 𝑛-Locally 𝐵)
Theorem	llytop 23366	A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ Top)
Theorem	nllytop 23367	A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top)
Theorem	llyi 23368*	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 23369*	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 23370*	Eliminate the neighborhood symbol from nllyi 23369. (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 23371	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 23372	A locally 𝐴 space is n-locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
⊢ Locally 𝐴 ⊆ 𝑛-Locally 𝐴
Theorem	llyss 23373	The "locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
⊢ (𝐴 ⊆ 𝐵 → Locally 𝐴 ⊆ Locally 𝐵)
Theorem	nllyss 23374	The "n-locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
⊢ (𝐴 ⊆ 𝐵 → 𝑛-Locally 𝐴 ⊆ 𝑛-Locally 𝐵)
Theorem	subislly 23375*	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 23376*	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 23377*	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 23378*	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 23379	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 23380	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 23381	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 23382	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 23383	Idempotence of the "n-locally" predicate, i.e. being "n-locally 𝐴 " is a local property. (Use loclly 23381 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 23384	A topology is locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
⊢ Locally Top = Top
Theorem	topnlly 23385	A topology is n-locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
⊢ 𝑛-Locally Top = Top
Theorem	hauslly 23386	A Hausdorff space is locally Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.)
⊢ (𝐽 ∈ Haus → 𝐽 ∈ Locally Haus)
Theorem	hausnlly 23387	A Hausdorff space is n-locally Hausdorff iff it is locally Hausdorff (both notions are thus referred to as "locally Hausdorff"). (Contributed by Mario Carneiro, 2-Mar-2015.)
⊢ (𝐽 ∈ 𝑛-Locally Haus ↔ 𝐽 ∈ Locally Haus)
Theorem	hausllycmp 23388	A compact Hausdorff space is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → 𝐽 ∈ 𝑛-Locally Comp)
Theorem	cldllycmp 23389	A closed subspace of a locally compact space is also locally compact. (The analogous result for open subspaces follows from the more general nllyrest 23380.) (Contributed by Mario Carneiro, 2-Mar-2015.)
⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽 ↾t 𝐴) ∈ 𝑛-Locally Comp)
Theorem	lly1stc 23390	First-countability is a local property (unlike second-countability). (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ Locally 1stω = 1stω
Theorem	dislly 23391*	The discrete space 𝒫 𝑋 is locally 𝐴 if and only if every singleton space has property 𝐴. (Contributed by Mario Carneiro, 20-Mar-2015.)
⊢ (𝑋 ∈ 𝑉 → (𝒫 𝑋 ∈ Locally 𝐴 ↔ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴))
Theorem	disllycmp 23392	A discrete space is locally compact. (Contributed by Mario Carneiro, 20-Mar-2015.)
⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Locally Comp)
Theorem	dis1stc 23393	A discrete space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ 1stω)
Theorem	hausmapdom 23394	If 𝑋 is a first-countable Hausdorff space, then the cardinality of the closure of a set 𝐴 is bounded by ℕ to the power 𝐴. In particular, a first-countable Hausdorff space with a dense subset 𝐴 has cardinality at most 𝐴↑ℕ, and a separable first-countable Hausdorff space has cardinality at most 𝒫 ℕ. (Compare hauspwpwdom 23882 to see a weaker result if the assumption of first-countability is omitted.) (Contributed by Mario Carneiro, 9-Apr-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ≼ (𝐴 ↑m ℕ))
Theorem	hauspwdom 23395	Simplify the cardinal 𝐴↑ℕ of hausmapdom 23394 to 𝒫 𝐵 = 2↑𝐵 when 𝐵 is an infinite cardinal greater than 𝐴. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) ∧ (𝐴 ≼ 𝒫 𝐵 ∧ ℕ ≼ 𝐵)) → ((cls‘𝐽)‘𝐴) ≼ 𝒫 𝐵)
12.1.16  Refinements
Syntax	cref 23396	Extend class definition to include the refinement relation.
class Ref
Syntax	cptfin 23397	Extend class definition to include the class of point-finite covers.
class PtFin
Syntax	clocfin 23398	Extend class definition to include the class of locally finite covers.
class LocFin
Definition	df-ref 23399*	Define the refinement relation. (Contributed by Jeff Hankins, 18-Jan-2010.)
⊢ Ref = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑦 = ∪ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)}
Definition	df-ptfin 23400*	Define "point-finite." (Contributed by Jeff Hankins, 21-Jan-2010.)
⊢ PtFin = {𝑥 ∣ ∀𝑦 ∈ ∪ 𝑥{𝑧 ∈ 𝑥 ∣ 𝑦 ∈ 𝑧} ∈ Fin}