P. 235 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23401-23500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cldmre 23401	The closed sets of a topology comprise a Moore system on the points of the topology. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘𝑋))
Theorem	mrccls 23402	Moore closure generalizes closure in a topology. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐹 = (mrCls‘(Clsd‘𝐽))    ⇒   ⊢ (𝐽 ∈ Top → (cls‘𝐽) = 𝐹)
Theorem	cls0 23403	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 23404	The interior of the empty set. (Contributed by NM, 2-Oct-2007.)
⊢ (𝐽 ∈ Top → ((int‘𝐽)‘∅) = ∅)
Theorem	isopn3i 23405	An open subset equals its own interior. (Contributed by Mario Carneiro, 30-Dec-2016.)
⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → ((int‘𝐽)‘𝑆) = 𝑆)
Theorem	elcls3 23406*	Membership in a closure in terms of the members of a basis. Theorem 6.5(b) of [Munkres] p. 95. (Contributed by NM, 26-Feb-2007.) (Revised by Mario Carneiro, 3-Sep-2015.)
⊢ (𝜑 → 𝐽 = (topGen‘𝐵))    &   ⊢ (𝜑 → 𝑋 = ∪ 𝐽)    &   ⊢ (𝜑 → 𝐵 ∈ TopBases)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝑃 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ 𝐵 (𝑃 ∈ 𝑥 → (𝑥 ∩ 𝑆) ≠ ∅)))
Theorem	opncldf1 23407*	A bijection useful for converting statements about open sets to statements about closed sets and vice versa. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐹 = (𝑢 ∈ 𝐽 ↦ (𝑋 ∖ 𝑢))    ⇒   ⊢ (𝐽 ∈ Top → (𝐹:𝐽–1-1-onto→(Clsd‘𝐽) ∧ ◡𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))))
Theorem	opncldf2 23408*	The values of the open-closed bijection. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐹 = (𝑢 ∈ 𝐽 ↦ (𝑋 ∖ 𝑢))    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝐹‘𝐴) = (𝑋 ∖ 𝐴))
Theorem	opncldf3 23409*	The values of the converse/inverse of the open-closed bijection. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐹 = (𝑢 ∈ 𝐽 ↦ (𝑋 ∖ 𝑢))    ⇒   ⊢ (𝐵 ∈ (Clsd‘𝐽) → (◡𝐹‘𝐵) = (𝑋 ∖ 𝐵))
Theorem	isclo 23410*	A set 𝐴 is clopen iff for every point 𝑥 in the space there is a neighborhood 𝑦 such that all the points in 𝑦 are in 𝐴 iff 𝑥 is. (Contributed by Mario Carneiro, 10-Mar-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
Theorem	isclo2 23411*	A set 𝐴 is clopen iff for every point 𝑥 in the space there is a neighborhood 𝑦 of 𝑥 which is either disjoint from 𝐴 or contained in 𝐴. (Contributed by Mario Carneiro, 7-Jul-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴))))
Theorem	discld 23412	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 23413	The closed sets of the topology {∅}. (Contributed by FL, 5-Jan-2009.)
⊢ (Clsd‘{∅}) = {∅}
Theorem	indiscld 23414	The closed sets of an indiscrete topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 14-Aug-2015.)
⊢ (Clsd‘{∅, 𝐴}) = {∅, 𝐴}
Theorem	mretopd 23415*	A Moore collection which is closed under finite unions called topological; such a collection is the closed sets of a canonically associated topology. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ (𝜑 → 𝑀 ∈ (Moore‘𝐵))    &   ⊢ (𝜑 → ∅ ∈ 𝑀)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀) → (𝑥 ∪ 𝑦) ∈ 𝑀)    &   ⊢ 𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ 𝑀}    ⇒   ⊢ (𝜑 → (𝐽 ∈ (TopOn‘𝐵) ∧ 𝑀 = (Clsd‘𝐽)))
Theorem	toponmre 23416	The topologies over a given base set form a Moore collection: the intersection of any family of them is a topology, including the empty (relative) intersection which gives the discrete topology distop 23318. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 5-May-2015.)
⊢ (𝐵 ∈ 𝑉 → (TopOn‘𝐵) ∈ (Moore‘𝒫 𝐵))
Theorem	cldmreon 23417	The closed sets of a topology over a set are a Moore collection over the same set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ (𝐽 ∈ (TopOn‘𝐵) → (Clsd‘𝐽) ∈ (Moore‘𝐵))
Theorem	iscldtop 23418*	A family is the closed sets of a topology iff it is a Moore collection and closed under finite union. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ (𝐾 ∈ (Clsd “ (TopOn‘𝐵)) ↔ (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾))
Theorem	mreclatdemoBAD 23419	The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclatBAD 18742. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 7377 update): This proof uses the old df-clat 18678 and references the required instance of mreclatBAD 18742 as a hypothesis. When mreclatBAD 18742 is corrected to become mreclat, delete this theorem and uncomment the mreclatdemo below.
⊢ (((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊))) ∈ (Moore‘∪ (TopOpen‘𝑊)) → (toInc‘((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊)))) ∈ CLat)    ⇒   ⊢ (𝑊 ∈ (TopSp ∩ LMod) → (toInc‘((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊)))) ∈ CLat)
12.1.5  Neighborhoods
Syntax	cnei 23420	Extend class notation with neighborhood relation for topologies.
class nei
Definition	df-nei 23421*	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 23422*	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 23423	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 23424	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 23425*	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 23426*	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 23427	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 23428*	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 23429	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 23430	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 23431*	Property of a neighborhood. (Contributed by NM, 12-Feb-2007.)
⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))
Theorem	neiss 23432	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 23433	A set is included in any of its neighborhoods. Generalization to subsets of elnei 23434. (Contributed by FL, 16-Nov-2006.)
⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑁)
Theorem	elnei 23434	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 23435	The empty set is not a neighborhood of a nonempty set. (Contributed by FL, 18-Sep-2007.)
⊢ ((𝐽 ∈ Top ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
Theorem	neips 23436*	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.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ ∀𝑝 ∈ 𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝})))
Theorem	opnneissb 23437	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 23438	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 23439	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	neindisj 23440	Any neighborhood of an element in the closure of a subset intersects the subset. Part of proof of Theorem 6.6 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑃 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃}))) → (𝑁 ∩ 𝑆) ≠ ∅)
Theorem	opnneiss 23441	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 23442	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	opnnei 23443*	A set is open iff it is a neighborhood of all of its points. (Contributed by Jeff Hankins, 15-Sep-2009.)
⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
Theorem	tpnei 23444	The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 23441. (Contributed by FL, 2-Oct-2006.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → (𝑆 ⊆ 𝑋 ↔ 𝑋 ∈ ((nei‘𝐽)‘𝑆)))
Theorem	neiuni 23445	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	neindisj2 23446*	A point 𝑃 belongs to the closure of a set 𝑆 iff every neighborhood of 𝑃 meets 𝑆. (Contributed by FL, 15-Sep-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑃})(𝑛 ∩ 𝑆) ≠ ∅))
Theorem	topssnei 23447	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 23448	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 23449	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 23450*	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 23451	The empty set is a neighborhood of itself. (Contributed by FL, 10-Dec-2006.)
⊢ (𝐽 ∈ Top → ∅ ∈ ((nei‘𝐽)‘∅))
Theorem	neipeltop 23452*	Lemma for neiptopreu 23456. (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 23453*	Lemma for neiptopreu 23456. (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 23454*	Lemma for neiptopreu 23456. (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 23455*	Lemma for neiptopreu 23456. (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 23456*	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 23433, innei 23448, elnei 23434 and neissex 23450, 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 23448 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 9406 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 23457	Extend class notation with the limit point function for topologies.
class limPt
Syntax	cperf 23458	Extend class notation with the class of all perfect spaces.
class Perf
Definition	df-lp 23459*	Define a function on topologies whose value is the set of limit points of the subsets of the base set. See lpval 23462. (Contributed by NM, 10-Feb-2007.)
⊢ limPt = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))}))
Definition	df-perf 23460	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 23461*	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 23462*	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 23463	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 23464	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 23465	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 23466	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 23467	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 23468*	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 23469*	The predicate "𝑃 is a limit point of 𝑆 " in terms of open sets. see islp2 23468, elcls 23396, islp 23463. (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 23470	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 23471	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 23472	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 23473	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 23474	Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋))
Theorem	isperf2 23475	Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))
Theorem	isperf3 23476*	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 23477	The limit points of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Perf → ((limPt‘𝐽)‘𝑋) = 𝑋)
Theorem	perfi 23478	Property of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Perf ∧ 𝑃 ∈ 𝑋) → ¬ {𝑃} ∈ 𝐽)
Theorem	perftop 23479	A perfect space is a topology. (Contributed by Mario Carneiro, 25-Dec-2016.)
⊢ (𝐽 ∈ Perf → 𝐽 ∈ Top)
12.1.7  Subspace topologies
Theorem	restrcl 23480	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 23481	A subspace topology basis is a basis. (Contributed by Mario Carneiro, 19-Mar-2015.)
⊢ (𝐵 ∈ TopBases → (𝐵 ↾t 𝐴) ∈ TopBases)
Theorem	tgrest 23482	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 23483	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 23484	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 23485	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 23486	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 23487	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 23488	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 23489	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 23490	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 23491	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 23492	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 23493	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 23494	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 23495*	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 23496	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 23497	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 23498	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 23499	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 23500	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)))