P. 230 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 22901-23000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	en2top 22901	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 22902	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 22903*	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 22904	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 22905*	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 22906	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 22907	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 22908	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 22909*	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 22910*	A version of bastop1 22909 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 22911	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 22912	The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 7761; an alternate proof uses indiscrete topologies (see indistop 22918) and the analogue of pwnex 7761 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7759). (Contributed by BJ, 2-May-2021.)
⊢ Top ∉ V
Theorem	distopon 22913	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 22914	The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ {∅} ∈ (TopOn‘∅)
Theorem	sn0top 22915	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 22916	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 22917	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 22918	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 22919	The base set of the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ ( I ‘𝐴) = ∪ {∅, 𝐴}
Theorem	fctop 22920*	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 22921*	The finite complement topology on a set 𝐴. Example 3 in [Munkres] p. 77. (This version of fctop 22920 requires the Axiom of Infinity.) (Contributed by FL, 20-Aug-2006.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≺ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))
Theorem	cctop 22922*	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 22923*	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 22924*	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 22925*	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 22926	The indiscrete topology on a set 𝐴 expressed as a topological space, using explicit structure component references. Compare with indistps 22927 and indistps2 22928. The advantage of this version is that the actual function for the structure is evident, and df-ndx 17163 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 17181 and df-tset 17252 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 22927 instead. (New usage is discouraged.) (Contributed by FL, 19-Jul-2006.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐾 = {⟨1, 𝐴⟩, ⟨9, {∅, 𝐴}⟩}    ⇒   ⊢ 𝐾 ∈ TopSp
Theorem	indistps 22927	The indiscrete topology on a set 𝐴 expressed as a topological space, using implicit structure indices. The advantage of this version over indistpsx 22926 is that it is independent of the indices of the component definitions df-base 17181 and df-tset 17252, and if they are changed in the future, this theorem will not be affected. The advantage over indistps2 22928 is that it is easy to eliminate the hypotheses with eqid 2728 and vtoclg 3540 to result in a closed theorem. Theorems indistpsALT 22929 and indistps2ALT 22931 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 22928	The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Compare with indistps 22927. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 22929 and indistps2ALT 22931 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 22929	The indiscrete topology on a set 𝐴 expressed as a topological space. Here we show how to derive the structural version indistps 22927 from the direct component assignment version indistps2 22928. (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 22930	Obsolete version of indistpsALT 22929 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 22927 from the direct component assignment version indistps2 22928. (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 22931	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 22928 from the structural version indistps 22927. (Contributed by NM, 24-Oct-2012.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (Base‘𝐾) = 𝐴    &   ⊢ (TopOpen‘𝐾) = {∅, 𝐴}    ⇒   ⊢ 𝐾 ∈ TopSp
Theorem	distps 22932	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 22933	Extend class notation with the set of closed sets of a topology.
class Clsd
Syntax	cnt 22934	Extend class notation with interior of a subset of a topology base set.
class int
Syntax	ccl 22935	Extend class notation with closure of a subset of a topology base set.
class cls
Definition	df-cld 22936*	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 22937*	Define a function on topologies whose value is the interior function on the subsets of the base set. See ntrval 22953. (Contributed by NM, 10-Sep-2006.)
⊢ int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∪ (𝑗 ∩ 𝒫 𝑥)))
Definition	df-cls 22938*	Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 22954. (Contributed by NM, 3-Oct-2006.)
⊢ cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∩ {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥 ⊆ 𝑦}))
Theorem	fncld 22939	The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ Clsd Fn Top
Theorem	cldval 22940*	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 22941*	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 22942*	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 22943	Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
Theorem	iscld 22944	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 22945	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 22946	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 22947	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 22948	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 22949	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 22950	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 22951	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 22952	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 22953	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 22954*	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 22955	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 22956*	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 22957	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 22958	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 22959	A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.)
⊢ (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆)
Theorem	incld 22960	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 22961*	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 22962*	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 22963	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 22964	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 22965	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 22966	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 22967	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 22968	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 22969	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 22970	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 22971	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 22972	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 22973	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 22974	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 22975	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 22976	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 22977	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 22978	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 22979	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 22980	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 22981	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 22982	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 22983	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 22984	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 22985	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 22986	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 22987	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 22988	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 22989	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 22990*	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 22991*	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 22992	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 22993	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 22994*	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 22995	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 22996	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 22997	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 22998	The interior of the empty set. (Contributed by NM, 2-Oct-2007.)
⊢ (𝐽 ∈ Top → ((int‘𝐽)‘∅) = ∅)
Theorem	isopn3i 22999	An open subset equals its own interior. (Contributed by Mario Carneiro, 30-Dec-2016.)
⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → ((int‘𝐽)‘𝑆) = 𝑆)
Theorem	elcls3 23000*	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) ≠ ∅)))