P. 231 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23001-23100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ntrss2 23001	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 23002	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	clsss3 23003	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	ntrss3 23004	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 23005	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	cmclsopn 23006	The complement of a closure is open. (Contributed by NM, 11-Sep-2006.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
Theorem	cmntrcld 23007	The complement of an interior is closed. (Contributed by NM, 1-Oct-2007.) (Proof shortened by OpenAI, 3-Jul-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ((int‘𝐽)‘𝑆)) ∈ (Clsd‘𝐽))
Theorem	iscld3 23008	A subset is closed iff it equals its own closure. (Contributed by NM, 2-Oct-2006.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑆) = 𝑆))
Theorem	iscld4 23009	A subset is closed iff it contains its own closure. (Contributed by NM, 31-Jan-2008.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑆) ⊆ 𝑆))
Theorem	isopn3 23010	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	clsidm 23011	The closure operation is idempotent. (Contributed by NM, 2-Oct-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘((cls‘𝐽)‘𝑆)) = ((cls‘𝐽)‘𝑆))
Theorem	ntridm 23012	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 23013	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 23014	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	0ntr 23015	A subset with an empty interior cannot cover a whole (nonempty) topology. (Contributed by NM, 12-Sep-2006.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (((𝐽 ∈ Top ∧ 𝑋 ≠ ∅) ∧ (𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → (𝑋 ∖ 𝑆) ≠ ∅)
Theorem	clsss2 23016	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	elcls 23017*	Membership in a closure. Theorem 6.5(a) of [Munkres] p. 95. (Contributed by NM, 22-Feb-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → (𝑥 ∩ 𝑆) ≠ ∅)))
Theorem	elcls2 23018*	Membership in a closure. (Contributed by NM, 5-Mar-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → (𝑥 ∩ 𝑆) ≠ ∅))))
Theorem	clsndisj 23019	Any open set containing a point that belongs to the closure of a subset intersects the subset. One direction of Theorem 6.5(a) of [Munkres] p. 95. (Contributed by NM, 26-Feb-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈)) → (𝑈 ∩ 𝑆) ≠ ∅)
Theorem	ntrcls0 23020	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 23021*	Two ways to say that a subset has an empty interior. (Contributed by NM, 3-Oct-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (((int‘𝐽)‘𝑆) = ∅ ↔ ∀𝑥 ∈ 𝐽 (𝑥 ⊆ 𝑆 → 𝑥 = ∅)))
Theorem	cldmre 23022	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 23023	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 23024	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 23025	The interior of the empty set. (Contributed by NM, 2-Oct-2007.)
⊢ (𝐽 ∈ Top → ((int‘𝐽)‘∅) = ∅)
Theorem	isopn3i 23026	An open subset equals its own interior. (Contributed by Mario Carneiro, 30-Dec-2016.)
⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → ((int‘𝐽)‘𝑆) = 𝑆)
Theorem	elcls3 23027*	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 23028*	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 23029*	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 23030*	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 23031*	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 23032*	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 23033	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 23034	The closed sets of the topology {∅}. (Contributed by FL, 5-Jan-2009.)
⊢ (Clsd‘{∅}) = {∅}
Theorem	indiscld 23035	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 23036*	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 23037	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 22939. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 5-May-2015.)
⊢ (𝐵 ∈ 𝑉 → (TopOn‘𝐵) ∈ (Moore‘𝒫 𝐵))
Theorem	cldmreon 23038	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 23039*	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 23040	The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclatBAD 18486. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 7315 update): This proof uses the old df-clat 18422 and references the required instance of mreclatBAD 18486 as a hypothesis. When mreclatBAD 18486 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 23041	Extend class notation with neighborhood relation for topologies.
class nei
Definition	df-nei 23042*	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 23043*	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 23044	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 23045	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 23046*	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 23047*	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 23048	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 23049*	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 23050	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 23051	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 23052*	Property of a neighborhood. (Contributed by NM, 12-Feb-2007.)
⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))
Theorem	neiss 23053	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 23054	A set is included in any of its neighborhoods. Generalization to subsets of elnei 23055. (Contributed by FL, 16-Nov-2006.)
⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑁)
Theorem	elnei 23055	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 23056	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 23057*	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 23058	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 23059	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 23060	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 23061	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 23062	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 23063	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 23064*	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 23065	The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 23062. (Contributed by FL, 2-Oct-2006.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → (𝑆 ⊆ 𝑋 ↔ 𝑋 ∈ ((nei‘𝐽)‘𝑆)))
Theorem	neiuni 23066	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 23067*	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 23068	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 23069	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 23070	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 23071*	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 23072	The empty set is a neighborhood of itself. (Contributed by FL, 10-Dec-2006.)
⊢ (𝐽 ∈ Top → ∅ ∈ ((nei‘𝐽)‘∅))
Theorem	neipeltop 23073*	Lemma for neiptopreu 23077. (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 23074*	Lemma for neiptopreu 23077. (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 23075*	Lemma for neiptopreu 23077. (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 23076*	Lemma for neiptopreu 23077. (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 23077*	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 23054, innei 23069, elnei 23055 and neissex 23071, 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 23069 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 9314 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 23078	Extend class notation with the limit point function for topologies.
class limPt
Syntax	cperf 23079	Extend class notation with the class of all perfect spaces.
class Perf
Definition	df-lp 23080*	Define a function on topologies whose value is the set of limit points of the subsets of the base set. See lpval 23083. (Contributed by NM, 10-Feb-2007.)
⊢ limPt = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))}))
Definition	df-perf 23081	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 23082*	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 23083*	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 23084	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 23085	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 23086	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 23087	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 23088	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 23089*	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 23090*	The predicate "𝑃 is a limit point of 𝑆 " in terms of open sets. see islp2 23089, elcls 23017, islp 23084. (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 23091	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 23092	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 23093	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 23094	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 23095	Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋))
Theorem	isperf2 23096	Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))
Theorem	isperf3 23097*	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 23098	The limit points of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Perf → ((limPt‘𝐽)‘𝑋) = 𝑋)
Theorem	perfi 23099	Property of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Perf ∧ 𝑃 ∈ 𝑋) → ¬ {𝑃} ∈ 𝐽)
Theorem	perftop 23100	A perfect space is a topology. (Contributed by Mario Carneiro, 25-Dec-2016.)
⊢ (𝐽 ∈ Perf → 𝐽 ∈ Top)