P. 214 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21301-21400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	neissex 21301*	For any neighborhood 𝑁 of 𝑆, there is a neighborhood 𝑥 of 𝑆 such that 𝑁 is a neighborhood of all subsets of 𝑥. Generalization to subsets of Property Viv of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.)
⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑥 ∈ ((nei‘𝐽)‘𝑆)∀𝑦(𝑦 ⊆ 𝑥 → 𝑁 ∈ ((nei‘𝐽)‘𝑦)))
Theorem	0nei 21302	The empty set is a neighborhood of itself. (Contributed by FL, 10-Dec-2006.)
⊢ (𝐽 ∈ Top → ∅ ∈ ((nei‘𝐽)‘∅))
Theorem	neipeltop 21303*	Lemma for neiptopreu 21307. (Contributed by Thierry Arnoux, 6-Jan-2018.)
⊢ 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)}    ⇒   ⊢ (𝐶 ∈ 𝐽 ↔ (𝐶 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝)))
Theorem	neiptopuni 21304*	Lemma for neiptopreu 21307. (Contributed by Thierry Arnoux, 6-Jan-2018.)
⊢ 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)}    &   ⊢ (𝜑 → 𝑁:𝑋⟶𝒫 𝒫 𝑋)    &   ⊢ ((((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑏 ∈ (𝑁‘𝑝))    &   ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝))    &   ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎)    &   ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑏 ∈ (𝑁‘𝑝)∀𝑞 ∈ 𝑏 𝑎 ∈ (𝑁‘𝑞))    &   ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))    ⇒   ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
Theorem	neiptoptop 21305*	Lemma for neiptopreu 21307. (Contributed by Thierry Arnoux, 7-Jan-2018.)
⊢ 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)}    &   ⊢ (𝜑 → 𝑁:𝑋⟶𝒫 𝒫 𝑋)    &   ⊢ ((((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑏 ∈ (𝑁‘𝑝))    &   ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝))    &   ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎)    &   ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑏 ∈ (𝑁‘𝑝)∀𝑞 ∈ 𝑏 𝑎 ∈ (𝑁‘𝑞))    &   ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))    ⇒   ⊢ (𝜑 → 𝐽 ∈ Top)
Theorem	neiptopnei 21306*	Lemma for neiptopreu 21307. (Contributed by Thierry Arnoux, 7-Jan-2018.)
⊢ 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)}    &   ⊢ (𝜑 → 𝑁:𝑋⟶𝒫 𝒫 𝑋)    &   ⊢ ((((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑏 ∈ (𝑁‘𝑝))    &   ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝))    &   ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎)    &   ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑏 ∈ (𝑁‘𝑝)∀𝑞 ∈ 𝑏 𝑎 ∈ (𝑁‘𝑞))    &   ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))    ⇒   ⊢ (𝜑 → 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝐽)‘{𝑝})))
Theorem	neiptopreu 21307*	If, to each element 𝑃 of a set 𝑋, we associate a set (𝑁‘𝑃) fulfilling Properties Vi, Vii, Viii and Property Viv of [BourbakiTop1] p. I.2. , corresponding to ssnei 21284, innei 21299, elnei 21285 and neissex 21301, then there is a unique topology 𝑗 such that for any point 𝑝, (𝑁‘𝑝) is the set of neighborhoods of 𝑝. Proposition 2 of [BourbakiTop1] p. I.3. This can be used to build a topology from a set of neighborhoods. Note that innei 21299 uses binary intersections whereas Property Vii mentions finite intersections (which includes the empty intersection of subsets of 𝑋, which is equal to 𝑋), so we add the hypothesis that 𝑋 is a neighborhood of all points. TODO: when df-fi 8585 includes the empty intersection, remove that extra hypothesis. (Contributed by Thierry Arnoux, 6-Jan-2018.)
⊢ 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)}    &   ⊢ (𝜑 → 𝑁:𝑋⟶𝒫 𝒫 𝑋)    &   ⊢ ((((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑏 ∈ (𝑁‘𝑝))    &   ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝))    &   ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎)    &   ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑏 ∈ (𝑁‘𝑝)∀𝑞 ∈ 𝑏 𝑎 ∈ (𝑁‘𝑞))    &   ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))    ⇒   ⊢ (𝜑 → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
12.1.6  Limit points and perfect sets
Syntax	clp 21308	Extend class notation with the limit point function for topologies.
class limPt
Syntax	cperf 21309	Extend class notation with the class of all perfect spaces.
class Perf
Definition	df-lp 21310*	Define a function on topologies whose value is the set of limit points of the subsets of the base set. See lpval 21313. (Contributed by NM, 10-Feb-2007.)
⊢ limPt = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))}))
Definition	df-perf 21311	Define the class of all perfect spaces. A perfect space is one for which every point in the set is a limit point of the whole space. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ Perf = {𝑗 ∈ Top ∣ ((limPt‘𝑗)‘∪ 𝑗) = ∪ 𝑗}
Theorem	lpfval 21312*	The limit point function on the subsets of a topology's base set. (Contributed by NM, 10-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → (limPt‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}))
Theorem	lpval 21313*	The set of limit points of a subset of the base set of a topology. Alternate definition of limit point in [Munkres] p. 97. (Contributed by NM, 10-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑆) = {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))})
Theorem	islp 21314	The predicate "the class 𝑃 is a limit point of 𝑆". (Contributed by NM, 10-Feb-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃}))))
Theorem	lpsscls 21315	The limit points of a subset are included in the subset's closure. (Contributed by NM, 26-Feb-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝑆))
Theorem	lpss 21316	The limit points of a subset are included in the base set. (Contributed by NM, 9-Nov-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑆) ⊆ 𝑋)
Theorem	lpdifsn 21317	𝑃 is a limit point of 𝑆 iff it is a limit point of 𝑆 ∖ {𝑃}. (Contributed by Mario Carneiro, 25-Dec-2016.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑃 ∈ ((limPt‘𝐽)‘(𝑆 ∖ {𝑃}))))
Theorem	lpss3 21318	Subset relationship for limit points. (Contributed by Mario Carneiro, 25-Dec-2016.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((limPt‘𝐽)‘𝑇) ⊆ ((limPt‘𝐽)‘𝑆))
Theorem	islp2 21319*	The predicate "𝑃 is a limit point of 𝑆 " in terms of neighborhoods. Definition of limit point in [Munkres] p. 97. Although Munkres uses open neighborhoods, it also works for our more general neighborhoods. (Contributed by NM, 26-Feb-2007.) (Proof shortened by Mario Carneiro, 25-Dec-2016.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅))
Theorem	islp3 21320*	The predicate "𝑃 is a limit point of 𝑆 " in terms of open sets. see islp2 21319, elcls 21247, islp 21314. (Contributed by FL, 31-Jul-2009.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → (𝑥 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)))
Theorem	maxlp 21321	A point is a limit point of the whole space iff the singleton of the point is not open. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ (𝑃 ∈ 𝑋 ∧ ¬ {𝑃} ∈ 𝐽)))
Theorem	clslp 21322	The closure of a subset of a topological space is the subset together with its limit points. Theorem 6.6 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))
Theorem	islpi 21323	A point belonging to a set's closure but not the set itself is a limit point. (Contributed by NM, 8-Nov-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑃 ∈ ((cls‘𝐽)‘𝑆) ∧ ¬ 𝑃 ∈ 𝑆)) → 𝑃 ∈ ((limPt‘𝐽)‘𝑆))
Theorem	cldlp 21324	A subset of a topological space is closed iff it contains all its limit points. Corollary 6.7 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((limPt‘𝐽)‘𝑆) ⊆ 𝑆))
Theorem	isperf 21325	Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋))
Theorem	isperf2 21326	Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))
Theorem	isperf3 21327*	A perfect space is a topology which has no open singletons. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽))
Theorem	perflp 21328	The limit points of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Perf → ((limPt‘𝐽)‘𝑋) = 𝑋)
Theorem	perfi 21329	Property of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Perf ∧ 𝑃 ∈ 𝑋) → ¬ {𝑃} ∈ 𝐽)
Theorem	perftop 21330	A perfect space is a topology. (Contributed by Mario Carneiro, 25-Dec-2016.)
⊢ (𝐽 ∈ Perf → 𝐽 ∈ Top)
12.1.7  Subspace topologies
Theorem	restrcl 21331	Reverse closure for the subspace topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
⊢ ((𝐽 ↾t 𝐴) ∈ Top → (𝐽 ∈ V ∧ 𝐴 ∈ V))
Theorem	restbas 21332	A subspace topology basis is a basis. 𝑌 is normally a subset of the base set of 𝐽. (Contributed by Mario Carneiro, 19-Mar-2015.)
⊢ (𝐵 ∈ TopBases → (𝐵 ↾t 𝐴) ∈ TopBases)
Theorem	tgrest 21333	A subspace can be generated by restricted sets from a basis for the original topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Proof shortened by Mario Carneiro, 30-Aug-2015.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (topGen‘(𝐵 ↾t 𝐴)) = ((topGen‘𝐵) ↾t 𝐴))
Theorem	resttop 21334	A subspace topology is a topology. Definition of subspace topology in [Munkres] p. 89. 𝐴 is normally a subset of the base set of 𝐽. (Contributed by FL, 15-Apr-2007.) (Revised by Mario Carneiro, 1-May-2015.)
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Top)
Theorem	resttopon 21335	A subspace topology is a topology on the base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
Theorem	restuni 21336	The underlying set of a subspace topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
Theorem	stoig 21337	The topological space built with a subspace topology. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), (𝐽 ↾t 𝐴)⟩} ∈ TopSp)
Theorem	restco 21338	Composition of subspaces. (Contributed by Mario Carneiro, 15-Dec-2013.) (Revised by Mario Carneiro, 1-May-2015.)
⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ((𝐽 ↾t 𝐴) ↾t 𝐵) = (𝐽 ↾t (𝐴 ∩ 𝐵)))
Theorem	restabs 21339	Equivalence of being a subspace of a subspace and being a subspace of the original. (Contributed by Jeff Hankins, 11-Jul-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.)
⊢ ((𝐽 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑇 ∧ 𝑇 ∈ 𝑊) → ((𝐽 ↾t 𝑇) ↾t 𝑆) = (𝐽 ↾t 𝑆))
Theorem	restin 21340	When the subspace region is not a subset of the base of the topology, the resulting set is the same as the subspace restricted to the base. (Contributed by Mario Carneiro, 15-Dec-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ 𝑋)))
Theorem	restuni2 21341	The underlying set of a subspace topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝑋) = ∪ (𝐽 ↾t 𝐴))
Theorem	resttopon2 21342	The underlying set of a subspace topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ (TopOn‘(𝐴 ∩ 𝑋)))
Theorem	rest0 21343	The subspace topology induced by the topology 𝐽 on the empty set. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 1-May-2015.)
⊢ (𝐽 ∈ Top → (𝐽 ↾t ∅) = {∅})
Theorem	restsn 21344	The only subspace topology induced by the topology {∅}. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
⊢ (𝐴 ∈ 𝑉 → ({∅} ↾t 𝐴) = {∅})
Theorem	restsn2 21345	The subspace topology induced by a singleton. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐽 ↾t {𝐴}) = 𝒫 {𝐴})
Theorem	restcld 21346*	A closed set of a subspace topology is a closed set of the original topology intersected with the subset. (Contributed by FL, 11-Jul-2009.) (Proof shortened by Mario Carneiro, 15-Dec-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘(𝐽 ↾t 𝑆)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆)))
Theorem	restcldi 21347	A closed set is closed in the subspace topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴)))
Theorem	restcldr 21348	A set which is closed in the subspace topology induced by a closed set is closed in the original topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴))) → 𝐵 ∈ (Clsd‘𝐽))
Theorem	restopnb 21349	If 𝐵 is an open subset of the subspace base set 𝐴, then any subset of 𝐵 is open iff it is open in 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐵)) → (𝐶 ∈ 𝐽 ↔ 𝐶 ∈ (𝐽 ↾t 𝐴)))
Theorem	ssrest 21350	If 𝐾 is a finer topology than 𝐽, then the subspace topologies induced by 𝐴 maintain this relationship. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
⊢ ((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) → (𝐽 ↾t 𝐴) ⊆ (𝐾 ↾t 𝐴))
Theorem	restopn2 21351	The if 𝐴 is open, then 𝐵 is open in 𝐴 iff it is an open subset of 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝐵 ∈ (𝐽 ↾t 𝐴) ↔ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴)))
Theorem	restdis 21352	A subspace of a discrete topology is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → (𝒫 𝐴 ↾t 𝐵) = 𝒫 𝐵)
Theorem	restfpw 21353	The restriction of the set of finite subsets of 𝐴 is the set of finite subsets of 𝐵. (Contributed by Mario Carneiro, 18-Sep-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → ((𝒫 𝐴 ∩ Fin) ↾t 𝐵) = (𝒫 𝐵 ∩ Fin))
Theorem	neitr 21354	The neighborhood of a trace is the trace of the neighborhood. (Contributed by Thierry Arnoux, 17-Jan-2018.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ((nei‘(𝐽 ↾t 𝐴))‘𝐵) = (((nei‘𝐽)‘𝐵) ↾t 𝐴))
Theorem	restcls 21355	A closure in a subspace topology. (Contributed by Jeff Hankins, 22-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (𝐽 ↾t 𝑌)    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
Theorem	restntr 21356	An interior in a subspace topology. Willard in General Topology says that there is no analogue of restcls 21355 for interiors. In some sense, that is true. (Contributed by Jeff Hankins, 23-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (𝐽 ↾t 𝑌)    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐾)‘𝑆) = (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌))
Theorem	restlp 21357	The limit points of a subset restrict naturally in a subspace. (Contributed by Mario Carneiro, 25-Dec-2016.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (𝐽 ↾t 𝑌)    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((limPt‘𝐾)‘𝑆) = (((limPt‘𝐽)‘𝑆) ∩ 𝑌))
Theorem	restperf 21358	Perfection of a subspace. Note that the term "perfect set" is reserved for closed sets which are perfect in the subspace topology. (Contributed by Mario Carneiro, 25-Dec-2016.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (𝐽 ↾t 𝑌)    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐾 ∈ Perf ↔ 𝑌 ⊆ ((limPt‘𝐽)‘𝑌)))
Theorem	perfopn 21359	An open subset of a perfect space is perfect. (Contributed by Mario Carneiro, 25-Dec-2016.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (𝐽 ↾t 𝑌)    ⇒   ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝐾 ∈ Perf)
Theorem	resstopn 21360	The topology of a restricted structure. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝐻 = (𝐾 ↾s 𝐴)    &   ⊢ 𝐽 = (TopOpen‘𝐾)    ⇒   ⊢ (𝐽 ↾t 𝐴) = (TopOpen‘𝐻)
Theorem	resstps 21361	A restricted topological space is a topological space. Note that this theorem would not be true if TopSp was defined directly in terms of the TopSet slot instead of the TopOpen derived function. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ ((𝐾 ∈ TopSp ∧ 𝐴 ∈ 𝑉) → (𝐾 ↾s 𝐴) ∈ TopSp)
12.1.8  Order topology
Theorem	ordtbaslem 21362*	Lemma for ordtbas 21366. In a total order, unbounded-above intervals are closed under intersection. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝑋 = dom 𝑅    &   ⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})    ⇒   ⊢ (𝑅 ∈ TosetRel → (fi‘𝐴) = 𝐴)
Theorem	ordtval 21363*	Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝑋 = dom 𝑅    &   ⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})    &   ⊢ 𝐵 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})    ⇒   ⊢ (𝑅 ∈ 𝑉 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (𝐴 ∪ 𝐵)))))
Theorem	ordtuni 21364*	Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝑋 = dom 𝑅    &   ⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})    &   ⊢ 𝐵 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝑋 = ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)))
Theorem	ordtbas2 21365*	Lemma for ordtbas 21366. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝑋 = dom 𝑅    &   ⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})    &   ⊢ 𝐵 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})    &   ⊢ 𝐶 = ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})    ⇒   ⊢ (𝑅 ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) ∪ 𝐶))
Theorem	ordtbas 21366*	In a total order, the finite intersections of the open rays generates the set of open intervals, but no more - these four collections form a subbasis for the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝑋 = dom 𝑅    &   ⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})    &   ⊢ 𝐵 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})    &   ⊢ 𝐶 = ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})    ⇒   ⊢ (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))) = (({𝑋} ∪ (𝐴 ∪ 𝐵)) ∪ 𝐶))
Theorem	ordttopon 21367	Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ (𝑅 ∈ 𝑉 → (ordTop‘𝑅) ∈ (TopOn‘𝑋))
Theorem	ordtopn1 21368*	An upward ray (𝑃, +∞) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ (ordTop‘𝑅))
Theorem	ordtopn2 21369*	A downward ray (-∞, 𝑃) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ (ordTop‘𝑅))
Theorem	ordtopn3 21370*	An open interval (𝐴, 𝐵) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ (¬ 𝑥𝑅𝐴 ∧ ¬ 𝐵𝑅𝑥)} ∈ (ordTop‘𝑅))
Theorem	ordtcld1 21371*	A downward ray (-∞, 𝑃] is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ 𝑥𝑅𝑃} ∈ (Clsd‘(ordTop‘𝑅)))
Theorem	ordtcld2 21372*	An upward ray [𝑃, +∞) is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ 𝑃𝑅𝑥} ∈ (Clsd‘(ordTop‘𝑅)))
Theorem	ordtcld3 21373*	A closed interval [𝐴, 𝐵] is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ (𝐴𝑅𝑥 ∧ 𝑥𝑅𝐵)} ∈ (Clsd‘(ordTop‘𝑅)))
Theorem	ordttop 21374	The order topology is a topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (𝑅 ∈ 𝑉 → (ordTop‘𝑅) ∈ Top)
Theorem	ordtcnv 21375	The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (𝑅 ∈ PosetRel → (ordTop‘◡𝑅) = (ordTop‘𝑅))
Theorem	ordtrest 21376	The subspace topology of an order topology is in general finer than the topology generated by the restricted order, but we do have inclusion in one direction. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
Theorem	ordtrest2lem 21377*	Lemma for ordtrest2 21378. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝑋 = dom 𝑅    &   ⊢ (𝜑 → 𝑅 ∈ TosetRel )    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑋)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝑋 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)} ⊆ 𝐴)    ⇒   ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
Theorem	ordtrest2 21378*	An interval-closed set 𝐴 in a total order has the same subspace topology as the restricted order topology. (An interval-closed set is the same thing as an open or half-open or closed interval in ℝ, but in other sets like ℚ there are interval-closed sets like (π, +∞) ∩ ℚ that are not intervals.) (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝑋 = dom 𝑅    &   ⊢ (𝜑 → 𝑅 ∈ TosetRel )    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑋)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝑋 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)} ⊆ 𝐴)    ⇒   ⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = ((ordTop‘𝑅) ↾t 𝐴))
Theorem	letopon 21379	The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ*)
Theorem	letop 21380	The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (ordTop‘ ≤ ) ∈ Top
Theorem	letopuni 21381	The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ ℝ* = ∪ (ordTop‘ ≤ )
Theorem	xrstopn 21382	The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ (ordTop‘ ≤ ) = (TopOpen‘ℝ*𝑠)
Theorem	xrstps 21383	The extended real number structure is a topological space. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ℝ*𝑠 ∈ TopSp
Theorem	leordtvallem1 21384*	Lemma for leordtval 21387. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞))    ⇒   ⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥})
Theorem	leordtvallem2 21385*	Lemma for leordtval 21387. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞))    &   ⊢ 𝐵 = ran (𝑥 ∈ ℝ* ↦ (-∞[,)𝑥))    ⇒   ⊢ 𝐵 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑥 ≤ 𝑦})
Theorem	leordtval2 21386	The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞))    &   ⊢ 𝐵 = ran (𝑥 ∈ ℝ* ↦ (-∞[,)𝑥))    ⇒   ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘(𝐴 ∪ 𝐵)))
Theorem	leordtval 21387	The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞))    &   ⊢ 𝐵 = ran (𝑥 ∈ ℝ* ↦ (-∞[,)𝑥))    &   ⊢ 𝐶 = ran (,)    ⇒   ⊢ (ordTop‘ ≤ ) = (topGen‘((𝐴 ∪ 𝐵) ∪ 𝐶))
Theorem	iccordt 21388	A closed interval is closed in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (𝐴[,]𝐵) ∈ (Clsd‘(ordTop‘ ≤ ))
Theorem	iocpnfordt 21389	An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (𝐴(,]+∞) ∈ (ordTop‘ ≤ )
Theorem	icomnfordt 21390	An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (-∞[,)𝐴) ∈ (ordTop‘ ≤ )
Theorem	iooordt 21391	An open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (𝐴(,)𝐵) ∈ (ordTop‘ ≤ )
Theorem	reordt 21392	The real numbers are an open set in the topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ ℝ ∈ (ordTop‘ ≤ )
Theorem	lecldbas 21393	The set of closed intervals forms a closed subbasis for the topology on the extended reals. Since our definition of a basis is in terms of open sets, we express this by showing that the complements of closed intervals form an open subbasis for the topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ ran [,] ↦ (ℝ* ∖ 𝑥))    ⇒   ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘ran 𝐹))
Theorem	pnfnei 21394*	A neighborhood of +∞ contains an unbounded interval based at a real number. Together with xrtgioo 22978 (which describes neighborhoods of ℝ) and mnfnei 21395, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 21391 and similar ensure that it has all the sets we want). (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ ((𝐴 ∈ (ordTop‘ ≤ ) ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
Theorem	mnfnei 21395*	A neighborhood of -∞ contains an unbounded interval based at a real number. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ ((𝐴 ∈ (ordTop‘ ≤ ) ∧ -∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴)
Theorem	ordtrestixx 21396*	The restriction of the less than order to an interval gives the same topology as the subspace topology. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝐴 ⊆ ℝ*    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥[,]𝑦) ⊆ 𝐴)    ⇒   ⊢ ((ordTop‘ ≤ ) ↾t 𝐴) = (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))
Theorem	ordtresticc 21397	The restriction of the less than order to a closed interval gives the same topology as the subspace topology. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ ((ordTop‘ ≤ ) ↾t (𝐴[,]𝐵)) = (ordTop‘( ≤ ∩ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))
12.1.9  Limits and continuity in topological spaces
Syntax	ccn 21398	Extend class notation with the class of continuous functions between topologies.
class Cn
Syntax	ccnp 21399	Extend class notation with the class of functions between topologies continuous at a given point.
class CnP
Syntax	clm 21400	Extend class notation with a function on topological spaces whose value is the convergence relation for limit sequences in the space.
class ⇝𝑡