P. 154 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15301-15400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	clsfval 15301*	The closure function on the subsets of a topology's base set. (Contributed by NM, 3-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦}))
Theorem	cldrcl 15302	Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
Theorem	iscld 15303	The predicate "the class 𝑆 is a closed set". (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽)))
Theorem	iscld2 15304	A subset of the underlying set of a topology is closed iff its complement is open. (Contributed by NM, 4-Oct-2006.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ 𝑆) ∈ 𝐽))
Theorem	cldss 15305	A closed set is a subset of the underlying set of a topology. (Contributed by NM, 5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ 𝑋)
Theorem	cldss2 15306	The set of closed sets is contained in the powerset of the base. (Contributed by Mario Carneiro, 6-Jan-2014.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋
Theorem	cldopn 15307	The complement of a closed set is open. (Contributed by NM, 5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝑆 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑆) ∈ 𝐽)
Theorem	difopn 15308	The difference of a closed set with an open set is open. (Contributed by Mario Carneiro, 6-Jan-2014.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∖ 𝐵) ∈ 𝐽)
Theorem	topcld 15309	The underlying set of a topology is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 3-Oct-2006.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
Theorem	ntrval 15310	The interior of a subset of a topology's base set is the union of all the open sets it includes. Definition of interior of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆))
Theorem	clsval 15311*	The closure of a subset of a topology's base set is the intersection of all the closed sets that include it. Definition of closure of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
Theorem	0cld 15312	The empty set is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 4-Oct-2006.)
⊢ (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽))
Theorem	uncld 15313	The union of two closed sets is closed. Equivalent to Theorem 6.1(3) of [Munkres] p. 93. (Contributed by NM, 5-Oct-2006.)
⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∪ 𝐵) ∈ (Clsd‘𝐽))
Theorem	cldcls 15314	A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.)
⊢ (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆)
Theorem	iuncld 15315*	A finite indexed union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.) (Revised by Jim Kingdon, 10-Mar-2023.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽))
Theorem	unicld 15316	A finite union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ (Clsd‘𝐽)) → ∪ 𝐴 ∈ (Clsd‘𝐽))
Theorem	ntropn 15317	The interior of a subset of a topology's underlying set is open. (Contributed by NM, 11-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
Theorem	clsss 15318	Subset relationship for closure. (Contributed by NM, 10-Feb-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((cls‘𝐽)‘𝑇) ⊆ ((cls‘𝐽)‘𝑆))
Theorem	ntrss 15319	Subset relationship for interior. (Contributed by NM, 3-Oct-2007.) (Revised by Jim Kingdon, 11-Mar-2023.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((int‘𝐽)‘𝑇) ⊆ ((int‘𝐽)‘𝑆))
Theorem	sscls 15320	A subset of a topology's underlying set is included in its closure. (Contributed by NM, 22-Feb-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
Theorem	ntrss2 15321	A subset includes its interior. (Contributed by NM, 3-Oct-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
Theorem	ssntr 15322	An open subset of a set is a subset of the set's interior. (Contributed by Jeff Hankins, 31-Aug-2009.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑂 ∈ 𝐽 ∧ 𝑂 ⊆ 𝑆)) → 𝑂 ⊆ ((int‘𝐽)‘𝑆))
Theorem	ntrss3 15323	The interior of a subset of a topological space is included in the space. (Contributed by NM, 1-Oct-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ⊆ 𝑋)
Theorem	ntrin 15324	A pairwise intersection of interiors is the interior of the intersection. This does not always hold for arbitrary intersections. (Contributed by Jeff Hankins, 31-Aug-2009.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝐴 ∩ 𝐵)) = (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)))
Theorem	isopn3 15325	A subset is open iff it equals its own interior. (Contributed by NM, 9-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆))
Theorem	ntridm 15326	The interior operation is idempotent. (Contributed by NM, 2-Oct-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘((int‘𝐽)‘𝑆)) = ((int‘𝐽)‘𝑆))
Theorem	clstop 15327	The closure of a topology's underlying set is the entire set. (Contributed by NM, 5-Oct-2007.) (Proof shortened by Jim Kingdon, 11-Mar-2023.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → ((cls‘𝐽)‘𝑋) = 𝑋)
Theorem	ntrtop 15328	The interior of a topology's underlying set is the entire set. (Contributed by NM, 12-Sep-2006.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → ((int‘𝐽)‘𝑋) = 𝑋)
Theorem	clsss2 15329	If a subset is included in a closed set, so is the subset's closure. (Contributed by NM, 22-Feb-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝑆) ⊆ 𝐶)
Theorem	clsss3 15330	The closure of a subset of a topological space is included in the space. (Contributed by NM, 26-Feb-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
Theorem	ntrcls0 15331	A subset whose closure has an empty interior also has an empty interior. (Contributed by NM, 4-Oct-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅) → ((int‘𝐽)‘𝑆) = ∅)
Theorem	ntreq0 15332*	Two ways to say that a subset has an empty interior. (Contributed by NM, 3-Oct-2007.) (Revised by Jim Kingdon, 11-Mar-2023.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (((int‘𝐽)‘𝑆) = ∅ ↔ ∀𝑥 ∈ 𝐽 (𝑥 ⊆ 𝑆 → 𝑥 = ∅)))
Theorem	cls0 15333	The closure of the empty set. (Contributed by NM, 2-Oct-2007.) (Proof shortened by Jim Kingdon, 12-Mar-2023.)
⊢ (𝐽 ∈ Top → ((cls‘𝐽)‘∅) = ∅)
Theorem	ntr0 15334	The interior of the empty set. (Contributed by NM, 2-Oct-2007.)
⊢ (𝐽 ∈ Top → ((int‘𝐽)‘∅) = ∅)
Theorem	isopn3i 15335	An open subset equals its own interior. (Contributed by Mario Carneiro, 30-Dec-2016.)
⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → ((int‘𝐽)‘𝑆) = 𝑆)
Theorem	discld 15336	The open sets of a discrete topology are closed and its closed sets are open. (Contributed by FL, 7-Jun-2007.) (Revised by Mario Carneiro, 7-Apr-2015.)
⊢ (𝐴 ∈ 𝑉 → (Clsd‘𝒫 𝐴) = 𝒫 𝐴)
Theorem	sn0cld 15337	The closed sets of the topology {∅}. (Contributed by FL, 5-Jan-2009.)
⊢ (Clsd‘{∅}) = {∅}
9.1.5  Neighborhoods
Syntax	cnei 15338	Extend class notation with neighborhood relation for topologies.
class nei
Definition	df-nei 15339*	Define a function on topologies whose value is a map from a subset to its neighborhoods. (Contributed by NM, 11-Feb-2007.)
⊢ nei = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∈ 𝒫 ∪ 𝑗 ∣ ∃𝑔 ∈ 𝑗 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦)}))
Theorem	neifval 15340*	Value of the neighborhood function on the subsets of the base set of a topology. (Contributed by NM, 11-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → (nei‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔 ∈ 𝐽 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣)}))
Theorem	neif 15341	The neighborhood function is a function from the set of the subsets of the base set of a topology. (Contributed by NM, 12-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → (nei‘𝐽) Fn 𝒫 𝑋)
Theorem	neiss2 15342	A set with a neighborhood is a subset of the base set of a topology. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by NM, 12-Feb-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋)
Theorem	neival 15343*	Value of the set of neighborhoods of a subset of the base set of a topology. (Contributed by NM, 11-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((nei‘𝐽)‘𝑆) = {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣)})
Theorem	isnei 15344*	The predicate "the class 𝑁 is a neighborhood of 𝑆". (Contributed by FL, 25-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
Theorem	neiint 15345	An intuitive definition of a neighborhood in terms of interior. (Contributed by Szymon Jaroszewicz, 18-Dec-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
Theorem	isneip 15346*	The predicate "the class 𝑁 is a neighborhood of point 𝑃". (Contributed by NM, 26-Feb-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑃 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
Theorem	neii1 15347	A neighborhood is included in the topology's base set. (Contributed by NM, 12-Feb-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 ⊆ 𝑋)
Theorem	neisspw 15348	The neighborhoods of any set are subsets of the base set. (Contributed by Stefan O'Rear, 6-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)
Theorem	neii2 15349*	Property of a neighborhood. (Contributed by NM, 12-Feb-2007.)
⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))
Theorem	neiss 15350	Any neighborhood of a set 𝑆 is also a neighborhood of any subset 𝑅 ⊆ 𝑆. Similar to Proposition 1 of [BourbakiTop1] p. I.2. (Contributed by FL, 25-Sep-2006.)
⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → 𝑁 ∈ ((nei‘𝐽)‘𝑅))
Theorem	ssnei 15351	A set is included in any of its neighborhoods. Generalization to subsets of elnei 15352. (Contributed by FL, 16-Nov-2006.)
⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑁)
Theorem	elnei 15352	A point belongs to any of its neighborhoods. Property Viii of [BourbakiTop1] p. I.3. (Contributed by FL, 28-Sep-2006.)
⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝐴 ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → 𝑃 ∈ 𝑁)
Theorem	0nnei 15353	The empty set is not a neighborhood of a nonempty set. (Contributed by FL, 18-Sep-2007.)
⊢ ((𝐽 ∈ Top ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
Theorem	neipsm 15354*	A neighborhood of a set is a neighborhood of every point in the set. Proposition 1 of [BourbakiTop1] p. I.2. (Contributed by FL, 16-Nov-2006.) (Revised by Jim Kingdon, 22-Mar-2023.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ ∃𝑥 𝑥 ∈ 𝑆) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ ∀𝑝 ∈ 𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝})))
Theorem	opnneissb 15355	An open set is a neighborhood of any of its subsets. (Contributed by FL, 2-Oct-2006.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑋) → (𝑆 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑆)))
Theorem	opnssneib 15356	Any superset of an open set is a neighborhood of it. (Contributed by NM, 14-Feb-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑆 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑆)))
Theorem	ssnei2 15357	Any subset 𝑀 of 𝑋 containing a neighborhood 𝑁 of a set 𝑆 is a neighborhood of this set. Generalization to subsets of Property Vi of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁 ⊆ 𝑀 ∧ 𝑀 ⊆ 𝑋)) → 𝑀 ∈ ((nei‘𝐽)‘𝑆))
Theorem	opnneiss 15358	An open set is a neighborhood of any of its subsets. (Contributed by NM, 13-Feb-2007.)
⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘𝑆))
Theorem	opnneip 15359	An open set is a neighborhood of any of its members. (Contributed by NM, 8-Mar-2007.)
⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑃 ∈ 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘{𝑃}))
Theorem	tpnei 15360	The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 15358. (Contributed by FL, 2-Oct-2006.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → (𝑆 ⊆ 𝑋 ↔ 𝑋 ∈ ((nei‘𝐽)‘𝑆)))
Theorem	neiuni 15361	The union of the neighborhoods of a set equals the topology's underlying set. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 9-Apr-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑋 = ∪ ((nei‘𝐽)‘𝑆))
Theorem	topssnei 15362	A finer topology has more neighborhoods. (Contributed by Mario Carneiro, 9-Apr-2015.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ 𝐽 ⊆ 𝐾) → ((nei‘𝐽)‘𝑆) ⊆ ((nei‘𝐾)‘𝑆))
Theorem	innei 15363	The intersection of two neighborhoods of a set is also a neighborhood of the set. Generalization to subsets of Property Vii of [BourbakiTop1] p. I.3 for binary intersections. (Contributed by FL, 28-Sep-2006.)
⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁 ∩ 𝑀) ∈ ((nei‘𝐽)‘𝑆))
Theorem	opnneiid 15364	Only an open set is a neighborhood of itself. (Contributed by FL, 2-Oct-2006.)
⊢ (𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑁) ↔ 𝑁 ∈ 𝐽))
Theorem	neissex 15365*	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 15366	The empty set is a neighborhood of itself. (Contributed by FL, 10-Dec-2006.)
⊢ (𝐽 ∈ Top → ∅ ∈ ((nei‘𝐽)‘∅))
9.1.6  Subspace topologies
Theorem	restrcl 15367	Reverse closure for the subspace topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Proof shortened by Jim Kingdon, 23-Mar-2023.)
⊢ ((𝐽 ↾t 𝐴) ∈ Top → (𝐽 ∈ V ∧ 𝐴 ∈ V))
Theorem	restbasg 15368	A subspace topology basis is a basis. (Contributed by Mario Carneiro, 19-Mar-2015.)
⊢ ((𝐵 ∈ TopBases ∧ 𝐴 ∈ 𝑉) → (𝐵 ↾t 𝐴) ∈ TopBases)
Theorem	tgrest 15369	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 15370	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 15371	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 15372	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 15373	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 15374	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 15375	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 15376	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 15377	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 15378	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 15379	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 15380	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	restopnb 15381	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 15382	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 15383	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 15384	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)
9.1.7  Limits and continuity in topological spaces
Syntax	ccn 15385	Extend class notation with the class of continuous functions between topologies.
class Cn
Syntax	ccnp 15386	Extend class notation with the class of functions between topologies continuous at a given point.
class CnP
Syntax	clm 15387	Extend class notation with a function on topological spaces whose value is the convergence relation for limit sequences in the space.
class ⇝𝑡
Definition	df-cn 15388*	Define a function on two topologies whose value is the set of continuous mappings from the first topology to the second. Based on definition of continuous function in [Munkres] p. 102. See iscn 15397 for the predicate form. (Contributed by NM, 17-Oct-2006.)
⊢ Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (∪ 𝑘 ↑𝑚 ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 (◡𝑓 “ 𝑦) ∈ 𝑗})
Definition	df-cnp 15389*	Define a function on two topologies whose value is the set of continuous mappings at a specified point in the first topology. Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 17-Oct-2006.)
⊢ CnP = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑥 ∈ ∪ 𝑗 ↦ {𝑓 ∈ (∪ 𝑘 ↑𝑚 ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}))
Definition	df-lm 15390*	Define a function on topologies whose value is the convergence relation for sequences into the given topological space. Although 𝑓 is typically a sequence (a function from an upperset of integers) with values in the topological space, it need not be. Note, however, that the limit property concerns only values at integers, so that the real-valued function (𝑥 ∈ ℝ ↦ (sin‘(π · 𝑥))) converges to zero (in the standard topology on the reals) with this definition. (Contributed by NM, 7-Sep-2006.)
⊢ ⇝𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))})
Theorem	lmrel 15391	The topological space convergence relation is a relation. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)
⊢ Rel (⇝𝑡‘𝐽)
Theorem	lmrcl 15392	Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.)
⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ Top)
Theorem	lmfval 15393*	The relation "sequence 𝑓 converges to point 𝑦 " in a metric space. (Contributed by NM, 7-Sep-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ (𝐽 ∈ (TopOn‘𝑋) → (⇝𝑡‘𝐽) = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋 ↑pm ℂ) ∧ 𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))})
Theorem	cnfval 15394*	The set of all continuous functions from topology 𝐽 to topology 𝐾. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 Cn 𝐾) = {𝑓 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽})
Theorem	cnpfval 15395*	The function mapping the points in a topology 𝐽 to the set of all functions from 𝐽 to topology 𝐾 continuous at that point. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 CnP 𝐾) = (𝑥 ∈ 𝑋 ↦ {𝑓 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑤 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑤 → ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 ∧ (𝑓 “ 𝑣) ⊆ 𝑤))}))
Theorem	cnovex 15396	The class of all continuous functions from a topology to another is a set. (Contributed by Jim Kingdon, 14-Dec-2023.)
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)
Theorem	iscn 15397*	The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾". Definition of continuous function in [Munkres] p. 102. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)))
Theorem	cnpval 15398*	The set of all functions from topology 𝐽 to topology 𝐾 that are continuous at a point 𝑃. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → ((𝐽 CnP 𝐾)‘𝑃) = {𝑓 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))})
Theorem	iscnp 15399*	The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾 at point 𝑃". Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))))
Theorem	iscn2 15400*	The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾". Definition of continuous function in [Munkres] p. 102. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)))