P. 228 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 22701-22800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	tgdom 22701	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 22702*	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 22703	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 22704	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 22705	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 22706	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 22707	{∅} is the only topology with one element. (Contributed by FL, 18-Aug-2008.)
⊢ (𝐽 ∈ Top → (𝐽 ≈ 1o ↔ 𝐽 = {∅}))
Theorem	en2top 22708	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 22709	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 22710*	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 22711	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 22712*	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 22713	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 22714	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 22715	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 22716*	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 22717*	A version of bastop1 22716 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 22718	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 22719	The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 7748; an alternate proof uses indiscrete topologies (see indistop 22725) and the analogue of pwnex 7748 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7746). (Contributed by BJ, 2-May-2021.)
⊢ Top ∉ V
Theorem	distopon 22720	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 22721	The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ {∅} ∈ (TopOn‘∅)
Theorem	sn0top 22722	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 22723	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 22724	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 22725	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 22726	The base set of the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ ( I ‘𝐴) = ∪ {∅, 𝐴}
Theorem	fctop 22727*	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 22728*	The finite complement topology on a set 𝐴. Example 3 in [Munkres] p. 77. (This version of fctop 22727 requires the Axiom of Infinity.) (Contributed by FL, 20-Aug-2006.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≺ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))
Theorem	cctop 22729*	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 22730*	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 22731*	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 22732*	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 22733	The indiscrete topology on a set 𝐴 expressed as a topological space, using explicit structure component references. Compare with indistps 22734 and indistps2 22735. The advantage of this version is that the actual function for the structure is evident, and df-ndx 17131 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 17149 and df-tset 17220 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 22734 instead. (New usage is discouraged.) (Contributed by FL, 19-Jul-2006.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐾 = {⟨1, 𝐴⟩, ⟨9, {∅, 𝐴}⟩}    ⇒   ⊢ 𝐾 ∈ TopSp
Theorem	indistps 22734	The indiscrete topology on a set 𝐴 expressed as a topological space, using implicit structure indices. The advantage of this version over indistpsx 22733 is that it is independent of the indices of the component definitions df-base 17149 and df-tset 17220, and if they are changed in the future, this theorem will not be affected. The advantage over indistps2 22735 is that it is easy to eliminate the hypotheses with eqid 2730 and vtoclg 3541 to result in a closed theorem. Theorems indistpsALT 22736 and indistps2ALT 22738 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 22735	The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Compare with indistps 22734. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 22736 and indistps2ALT 22738 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 22736	The indiscrete topology on a set 𝐴 expressed as a topological space. Here we show how to derive the structural version indistps 22734 from the direct component assignment version indistps2 22735. (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	indistpsALTOLD 22737	Obsolete version of indistpsALT 22736 as of 31-Oct-2024. The indiscrete topology on a set 𝐴 expressed as a topological space. Here we show how to derive the structural version indistps 22734 from the direct component assignment version indistps2 22735. (Contributed by NM, 24-Oct-2012.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), {∅, 𝐴}⟩}    ⇒   ⊢ 𝐾 ∈ TopSp
Theorem	indistps2ALT 22738	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 22735 from the structural version indistps 22734. (Contributed by NM, 24-Oct-2012.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (Base‘𝐾) = 𝐴    &   ⊢ (TopOpen‘𝐾) = {∅, 𝐴}    ⇒   ⊢ 𝐾 ∈ TopSp
Theorem	distps 22739	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 22740	Extend class notation with the set of closed sets of a topology.
class Clsd
Syntax	cnt 22741	Extend class notation with interior of a subset of a topology base set.
class int
Syntax	ccl 22742	Extend class notation with closure of a subset of a topology base set.
class cls
Definition	df-cld 22743*	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 22744*	Define a function on topologies whose value is the interior function on the subsets of the base set. See ntrval 22760. (Contributed by NM, 10-Sep-2006.)
⊢ int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∪ (𝑗 ∩ 𝒫 𝑥)))
Definition	df-cls 22745*	Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 22761. (Contributed by NM, 3-Oct-2006.)
⊢ cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∩ {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥 ⊆ 𝑦}))
Theorem	fncld 22746	The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ Clsd Fn Top
Theorem	cldval 22747*	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 22748*	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 22749*	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 22750	Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
Theorem	iscld 22751	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 22752	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 22753	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 22754	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 22755	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 22756	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 22757	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 22758	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 22759	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 22760	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 22761*	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 22762	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 22763*	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 22764	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 22765	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 22766	A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.)
⊢ (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆)
Theorem	incld 22767	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 22768*	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 22769*	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 22770	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 22771	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 22772	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 22773	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 22774	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 22775	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 22776	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 22777	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 22778	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 22779	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 22780	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 22781	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 22782	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 22783	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 22784	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 22785	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 22786	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 22787	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 22788	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 22789	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 22790	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 22791	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 22792	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 22793	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 22794	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 22795	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 22796	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 22797*	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 22798*	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 22799	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 22800	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) = ∅)