P. 232 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23101-23200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	tg2 23101*	Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
⊢ ((𝐴 ∈ (topGen‘𝐵) ∧ 𝐶 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 (𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))
Theorem	bastg 23102	A member of a basis is a subset of the topology it generates. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
⊢ (𝐵 ∈ 𝑉 → 𝐵 ⊆ (topGen‘𝐵))
Theorem	unitg 23103	The topology generated by a basis 𝐵 is a topology on ∪ 𝐵. Importantly, this theorem means that we don't have to specify separately the base set for the topological space generated by a basis. In other words, any member of the class TopBases completely specifies the basis it corresponds to. (Contributed by NM, 16-Jul-2006.) (Proof shortened by OpenAI, 30-Mar-2020.)
⊢ (𝐵 ∈ 𝑉 → ∪ (topGen‘𝐵) = ∪ 𝐵)
Theorem	tgss 23104	Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)
⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶))
Theorem	tgcl 23105	Show that a basis generates a topology. Remark in [Munkres] p. 79. (Contributed by NM, 17-Jul-2006.)
⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top)
Theorem	tgclb 23106	The property tgcl 23105 can be reversed: if the topology generated by 𝐵 is actually a topology, then 𝐵 must be a topological basis. This yields an alternative definition of TopBases. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ (𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top)
Theorem	tgtopon 23107	A basis generates a topology on ∪ 𝐵. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ (TopOn‘∪ 𝐵))
Theorem	topbas 23108	A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
⊢ (𝐽 ∈ Top → 𝐽 ∈ TopBases)
Theorem	tgtop 23109	A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
Theorem	eltop 23110	Membership in a topology, expressed without quantifiers. (Contributed by NM, 19-Jul-2006.)
⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ 𝐴 ⊆ ∪ (𝐽 ∩ 𝒫 𝐴)))
Theorem	eltop2 23111*	Membership in a topology. (Contributed by NM, 19-Jul-2006.)
⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))
Theorem	eltop3 23112*	Membership in a topology. (Contributed by NM, 19-Jul-2006.)
⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∃𝑥(𝑥 ⊆ 𝐽 ∧ 𝐴 = ∪ 𝑥)))
Theorem	fibas 23113	A collection of finite intersections is a basis. The initial set is a subbasis for the topology. (Contributed by Jeff Hankins, 25-Aug-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
⊢ (fi‘𝐴) ∈ TopBases
Theorem	tgdom 23114	A space has no more open sets than subsets of a basis. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 9-Apr-2015.)
⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ≼ 𝒫 𝐵)
Theorem	tgiun 23115*	The indexed union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐶 ∈ (topGen‘𝐵))
Theorem	tgidm 23116	The topology generator function is idempotent. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)
⊢ (𝐵 ∈ 𝑉 → (topGen‘(topGen‘𝐵)) = (topGen‘𝐵))
Theorem	bastop 23117	Two ways to express that a basis is a topology. (Contributed by NM, 18-Jul-2006.)
⊢ (𝐵 ∈ TopBases → (𝐵 ∈ Top ↔ (topGen‘𝐵) = 𝐵))
Theorem	tgtop11 23118	The topology generation function is one-to-one when applied to completed topologies. (Contributed by NM, 18-Jul-2006.)
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ (topGen‘𝐽) = (topGen‘𝐾)) → 𝐽 = 𝐾)
Theorem	0top 23119	The singleton of the empty set is the only topology possible for an empty underlying set. (Contributed by NM, 9-Sep-2006.)
⊢ (𝐽 ∈ Top → (∪ 𝐽 = ∅ ↔ 𝐽 = {∅}))
Theorem	en1top 23120	{∅} is the only topology with one element. (Contributed by FL, 18-Aug-2008.)
⊢ (𝐽 ∈ Top → (𝐽 ≈ 1o ↔ 𝐽 = {∅}))
Theorem	en2top 23121	If a topology has two elements, it is the indiscrete topology. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ≈ 2o ↔ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)))
Theorem	tgss3 23122	A criterion for determining whether one topology is finer than another. Lemma 2.2 of [Munkres] p. 80 using abbreviations. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))
Theorem	tgss2 23123*	A criterion for determining whether one topology is finer than another, based on a comparison of their bases. Lemma 2.2 of [Munkres] p. 80. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ ∀𝑥 ∈ ∪ 𝐵∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
Theorem	basgen 23124	Given a topology 𝐽, show that a subset 𝐵 satisfying the third antecedent is a basis for it. Lemma 2.3 of [Munkres] p. 81 using abbreviations. (Contributed by NM, 22-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)
⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽 ∧ 𝐽 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = 𝐽)
Theorem	basgen2 23125*	Given a topology 𝐽, show that a subset 𝐵 satisfying the third antecedent is a basis for it. Lemma 2.3 of [Munkres] p. 81. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽 ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)) → (topGen‘𝐵) = 𝐽)
Theorem	2basgen 23126	Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 2-Sep-2015.)
⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = (topGen‘𝐶))
Theorem	tgfiss 23127	If a subbase is included into a topology, so is the generated topology. (Contributed by FL, 20-Apr-2012.) (Proof shortened by Mario Carneiro, 10-Jan-2015.)
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → (topGen‘(fi‘𝐴)) ⊆ 𝐽)
Theorem	tgdif0 23128	A generated topology is not affected by the addition or removal of the empty set from the base. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵)
Theorem	bastop1 23129*	A subset of a topology is a basis for the topology iff every member of the topology is a union of members of the basis. We use the idiom "(topGen‘𝐵) = 𝐽 " to express "𝐵 is a basis for topology 𝐽 " since we do not have a separate notation for this. Definition 15.35 of [Schechter] p. 428. (Contributed by NM, 2-Feb-2008.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥 ∈ 𝐽 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
Theorem	bastop2 23130*	A version of bastop1 23129 that doesn't have 𝐵 ⊆ 𝐽 in the antecedent. (Contributed by NM, 3-Feb-2008.)
⊢ (𝐽 ∈ Top → ((topGen‘𝐵) = 𝐽 ↔ (𝐵 ⊆ 𝐽 ∧ ∀𝑥 ∈ 𝐽 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))))
12.1.3  Examples of topologies
Theorem	distop 23131	The discrete topology on a set 𝐴. Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 17-Jul-2006.) (Revised by Mario Carneiro, 19-Mar-2015.)
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
Theorem	topnex 23132	The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 7757; an alternate proof uses indiscrete topologies (see indistop 23138) and the analogue of pwnex 7757 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7755). (Contributed by BJ, 2-May-2021.)
⊢ Top ∉ V
Theorem	distopon 23133	The discrete topology on a set 𝐴, with base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
Theorem	sn0topon 23134	The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ {∅} ∈ (TopOn‘∅)
Theorem	sn0top 23135	The singleton of the empty set is a topology. (Contributed by Stefan Allan, 3-Mar-2006.) (Proof shortened by Mario Carneiro, 13-Aug-2015.)
⊢ {∅} ∈ Top
Theorem	indislem 23136	A lemma to eliminate some sethood hypotheses when dealing with the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}
Theorem	indistopon 23137	The indiscrete topology on a set 𝐴. Part of Example 2 in [Munkres] p. 77. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐴 ∈ 𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))
Theorem	indistop 23138	The indiscrete topology on a set 𝐴. Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 16-Jul-2006.) (Revised by Stefan Allan, 6-Nov-2008.) (Revised by Mario Carneiro, 13-Aug-2015.)
⊢ {∅, 𝐴} ∈ Top
Theorem	indisuni 23139	The base set of the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ ( I ‘𝐴) = ∪ {∅, 𝐴}
Theorem	fctop 23140*	The finite complement topology on a set 𝐴. Example 3 in [Munkres] p. 77. (Contributed by FL, 15-Aug-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ∈ Fin ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))
Theorem	fctop2 23141*	The finite complement topology on a set 𝐴. Example 3 in [Munkres] p. 77. (This version of fctop 23140 requires the Axiom of Infinity.) (Contributed by FL, 20-Aug-2006.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≺ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))
Theorem	cctop 23142*	The countable complement topology on a set 𝐴. Example 4 in [Munkres] p. 77. (Contributed by FL, 23-Aug-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))
Theorem	ppttop 23143*	The particular point topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))
Theorem	pptbas 23144*	The particular point topology is generated by a basis consisting of pairs {𝑥, 𝑃} for each 𝑥 ∈ 𝐴. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} = (topGen‘ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃})))
Theorem	epttop 23145*	The excluded point topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ∈ (TopOn‘𝐴))
Theorem	indistpsx 23146	The indiscrete topology on a set 𝐴 expressed as a topological space, using explicit structure component references. Compare with indistps 23147 and indistps2 23148. The advantage of this version is that the actual function for the structure is evident, and df-ndx 17253 is not needed, nor any other special definition outside of basic set theory. The disadvantage is that if the indices of the component definitions df-base 17269 and df-tset 17328 are changed in the future, this theorem will also have to be changed. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use indistps 23147 instead. (New usage is discouraged.) (Contributed by FL, 19-Jul-2006.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐾 = {⟨1, 𝐴⟩, ⟨9, {∅, 𝐴}⟩}    ⇒   ⊢ 𝐾 ∈ TopSp
Theorem	indistps 23147	The indiscrete topology on a set 𝐴 expressed as a topological space, using implicit structure indices. The advantage of this version over indistpsx 23146 is that it is independent of the indices of the component definitions df-base 17269 and df-tset 17328, and if they are changed in the future, this theorem will not be affected. The advantage over indistps2 23148 is that it is easy to eliminate the hypotheses with eqid 2761 and vtoclg 3521 to result in a closed theorem. Theorems indistpsALT 23149 and indistps2ALT 23150 show that the two forms can be derived from each other. (Contributed by FL, 19-Jul-2006.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), {∅, 𝐴}⟩}    ⇒   ⊢ 𝐾 ∈ TopSp
Theorem	indistps2 23148	The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Compare with indistps 23147. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 23149 and indistps2ALT 23150 show that the two forms can be derived from each other. (Contributed by NM, 24-Oct-2012.)
⊢ (Base‘𝐾) = 𝐴    &   ⊢ (TopOpen‘𝐾) = {∅, 𝐴}    ⇒   ⊢ 𝐾 ∈ TopSp
Theorem	indistpsALT 23149	The indiscrete topology on a set 𝐴 expressed as a topological space. Here we show how to derive the structural version indistps 23147 from the direct component assignment version indistps2 23148. (Contributed by NM, 24-Oct-2012.) (Revised by AV, 31-Oct-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), {∅, 𝐴}⟩}    ⇒   ⊢ 𝐾 ∈ TopSp
Theorem	indistps2ALT 23150	The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Here we show how to derive the direct component assignment version indistps2 23148 from the structural version indistps 23147. (Contributed by NM, 24-Oct-2012.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (Base‘𝐾) = 𝐴    &   ⊢ (TopOpen‘𝐾) = {∅, 𝐴}    ⇒   ⊢ 𝐾 ∈ TopSp
Theorem	distps 23151	The discrete topology on a set 𝐴 expressed as a topological space. (Contributed by FL, 20-Aug-2006.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), 𝒫 𝐴⟩}    ⇒   ⊢ 𝐾 ∈ TopSp
12.1.4  Closure and interior
Syntax	ccld 23152	Extend class notation with the set of closed sets of a topology.
class Clsd
Syntax	cnt 23153	Extend class notation with interior of a subset of a topology base set.
class int
Syntax	ccl 23154	Extend class notation with closure of a subset of a topology base set.
class cls
Definition	df-cld 23155*	Define a function on topologies whose value is the set of closed sets of the topology. (Contributed by NM, 2-Oct-2006.)
⊢ Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗})
Definition	df-ntr 23156*	Define a function on topologies whose value is the interior function on the subsets of the base set. See ntrval 23172. (Contributed by NM, 10-Sep-2006.)
⊢ int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∪ (𝑗 ∩ 𝒫 𝑥)))
Definition	df-cls 23157*	Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 23173. (Contributed by NM, 3-Oct-2006.)
⊢ cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∩ {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥 ⊆ 𝑦}))
Theorem	fncld 23158	The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ Clsd Fn Top
Theorem	cldval 23159*	The set of closed sets of a topology. (Note that the set of open sets is just the topology itself, so we don't have a separate definition.) (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽})
Theorem	ntrfval 23160*	The interior function on the subsets of a topology's base set. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)))
Theorem	clsfval 23161*	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 23162	Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
Theorem	iscld 23163	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 23164	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 23165	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 23166	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 23167	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	isopn2 23168	A subset of the underlying set of a topology is open iff its complement is closed. (Contributed by NM, 4-Oct-2006.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 ↔ (𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽)))
Theorem	opncld 23169	The complement of an open set is closed. (Contributed by NM, 6-Oct-2006.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → (𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽))
Theorem	difopn 23170	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 23171	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 23172	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 23173*	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 23174	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	iincld 23175*	The indexed intersection of a collection 𝐵(𝑥) of closed sets is closed. Theorem 6.1(2) of [Munkres] p. 93. (Contributed by NM, 5-Oct-2006.) (Revised by Mario Carneiro, 3-Sep-2015.)
⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽))
Theorem	intcld 23176	The intersection of a set of closed sets is closed. (Contributed by NM, 5-Oct-2006.)
⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ (Clsd‘𝐽)) → ∩ 𝐴 ∈ (Clsd‘𝐽))
Theorem	uncld 23177	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 23178	A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.)
⊢ (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆)
Theorem	incld 23179	The intersection of two closed sets is closed. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ 𝐵) ∈ (Clsd‘𝐽))
Theorem	riincld 23180*	An indexed relative intersection of closed sets is closed. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) ∈ (Clsd‘𝐽))
Theorem	iuncld 23181*	A finite indexed union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽))
Theorem	unicld 23182	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	clscld 23183	The closure of a subset of a topology's underlying set is closed. (Contributed by NM, 4-Oct-2006.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
Theorem	clsf 23184	The closure function is a function from subsets of the base to closed sets. (Contributed by Mario Carneiro, 11-Apr-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽))
Theorem	ntropn 23185	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	clsval2 23186	Express closure in terms of interior. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ 𝑆))))
Theorem	ntrval2 23187	Interior expressed in terms of closure. (Contributed by NM, 1-Oct-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑆))))
Theorem	ntrdif 23188	An interior of a complement is the complement of the closure. This set is also known as the exterior of 𝐴. (Contributed by Jeff Hankins, 31-Aug-2009.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘𝐴)))
Theorem	clsdif 23189	A closure of a complement is the complement of the interior. (Contributed by Jeff Hankins, 31-Aug-2009.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((int‘𝐽)‘𝐴)))
Theorem	clsss 23190	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 23191	Subset relationship for interior. (Contributed by NM, 3-Oct-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((int‘𝐽)‘𝑇) ⊆ ((int‘𝐽)‘𝑆))
Theorem	sscls 23192	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 23193	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 23194	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 23195	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 23196	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 23197	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 23198	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 23199	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 23200	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))