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Theorem List for Metamath Proof Explorer - 20801-20900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisbasis3g 20801* Express the predicate "𝐵 is a basis for a topology." Definition of basis in [Munkres] p. 78. (Contributed by NM, 17-Jul-2006.)
(𝐵𝐶 → (𝐵 ∈ TopBases ↔ (∀𝑥𝐵 𝑥 𝐵 ∧ ∀𝑥 𝐵𝑦𝐵 𝑥𝑦 ∧ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))))

Theorembasis1 20802 Property of a basis. (Contributed by NM, 16-Jul-2006.)
((𝐵 ∈ TopBases ∧ 𝐶𝐵𝐷𝐵) → (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷)))

Theorembasis2 20803* Property of a basis. (Contributed by NM, 17-Jul-2006.)
(((𝐵 ∈ TopBases ∧ 𝐶𝐵) ∧ (𝐷𝐵𝐴 ∈ (𝐶𝐷))) → ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷)))

Theoremfiinbas 20804* If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)

Theorembasdif0 20805 A basis is not affected by the addition or removal of the empty set. (Contributed by Mario Carneiro, 28-Aug-2015.)
((𝐵 ∖ {∅}) ∈ TopBases ↔ 𝐵 ∈ TopBases)

Theorembaspartn 20806* A disjoint system of sets is a basis for a topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝑃𝑉 ∧ ∀𝑥𝑃𝑦𝑃 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)) → 𝑃 ∈ TopBases)

Theoremtgval 20807* The topology generated by a basis. See also tgval2 20808 and tgval3 20815. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})

Theoremtgval2 20808* Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 20821) that (topGen‘𝐵) is indeed a topology (on 𝐵, see unitg 20819). See also tgval 20807 and tgval3 20815. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (topGen‘𝐵) = {𝑥 ∣ (𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))})

Theoremeltg 20809 Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))

Theoremeltg2 20810* Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ (𝐴 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝑦𝐴))))

Theoremeltg2b 20811* Membership in a topology generated by a basis. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝑦𝐴)))

Theoremeltg4i 20812 An open set in a topology generated by a basis is the union of all basic open sets contained in it. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝐴 ∈ (topGen‘𝐵) → 𝐴 = (𝐵 ∩ 𝒫 𝐴))

Theoremeltg3i 20813 The union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
((𝐵𝑉𝐴𝐵) → 𝐴 ∈ (topGen‘𝐵))

Theoremeltg3 20814* Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Mario Carneiro, 30-Aug-2015.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥𝐵𝐴 = 𝑥)))

Theoremtgval3 20815* Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. See also tgval 20807 and tgval2 20808. (Contributed by NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.)
(𝐵𝑉 → (topGen‘𝐵) = {𝑥 ∣ ∃𝑦(𝑦𝐵𝑥 = 𝑦)})

Theoremtg1 20816 Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
(𝐴 ∈ (topGen‘𝐵) → 𝐴 𝐵)

Theoremtg2 20817* Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
((𝐴 ∈ (topGen‘𝐵) ∧ 𝐶𝐴) → ∃𝑥𝐵 (𝐶𝑥𝑥𝐴))

Theorembastg 20818 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.)
(𝐵𝑉𝐵 ⊆ (topGen‘𝐵))

Theoremunitg 20819 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.)
(𝐵𝑉 (topGen‘𝐵) = 𝐵)

Theoremtgss 20820 Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)
((𝐶𝑉𝐵𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶))

Theoremtgcl 20821 Show that a basis generates a topology. Remark in [Munkres] p. 79. (Contributed by NM, 17-Jul-2006.)
(𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top)

Theoremtgclb 20822 The property tgcl 20821 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)

Theoremtgtopon 20823 A basis generates a topology on 𝐵. (Contributed by Mario Carneiro, 14-Aug-2015.)
(𝐵 ∈ TopBases → (topGen‘𝐵) ∈ (TopOn‘ 𝐵))

Theoremtopbas 20824 A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
(𝐽 ∈ Top → 𝐽 ∈ TopBases)

Theoremtgtop 20825 A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
(𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)

Theoremeltop 20826 Membership in a topology, expressed without quantifiers. (Contributed by NM, 19-Jul-2006.)
(𝐽 ∈ Top → (𝐴𝐽𝐴 (𝐽 ∩ 𝒫 𝐴)))

Theoremeltop2 20827* Membership in a topology. (Contributed by NM, 19-Jul-2006.)
(𝐽 ∈ Top → (𝐴𝐽 ↔ ∀𝑥𝐴𝑦𝐽 (𝑥𝑦𝑦𝐴)))

Theoremeltop3 20828* Membership in a topology. (Contributed by NM, 19-Jul-2006.)
(𝐽 ∈ Top → (𝐴𝐽 ↔ ∃𝑥(𝑥𝐽𝐴 = 𝑥)))

Theoremfibas 20829 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

Theoremtgdom 20830 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.)
(𝐵𝑉 → (topGen‘𝐵) ≼ 𝒫 𝐵)

Theoremtgiun 20831* The indexed union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
((𝐵𝑉 ∧ ∀𝑥𝐴 𝐶𝐵) → 𝑥𝐴 𝐶 ∈ (topGen‘𝐵))

Theoremtgidm 20832 The topology generator function is idempotent. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)
(𝐵𝑉 → (topGen‘(topGen‘𝐵)) = (topGen‘𝐵))

Theorembastop 20833 Two ways to express that a basis is a topology. (Contributed by NM, 18-Jul-2006.)
(𝐵 ∈ TopBases → (𝐵 ∈ Top ↔ (topGen‘𝐵) = 𝐵))

Theoremtgtop11 20834 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)

Theorem0top 20835 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 = {∅}))

Theoremen1top 20836 {∅} is the only topology with one element. (Contributed by FL, 18-Aug-2008.)
(𝐽 ∈ Top → (𝐽 ≈ 1𝑜𝐽 = {∅}))

Theoremen2top 20837 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 ≠ ∅)))

Theoremtgss3 20838 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.)
((𝐵𝑉𝐶𝑊) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))

Theoremtgss2 20839* 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.)
((𝐵𝑉 𝐵 = 𝐶) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ ∀𝑥 𝐵𝑦𝐵 (𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))

Theorembasgen 20840 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 ∧ 𝐵𝐽𝐽 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = 𝐽)

Theorembasgen2 20841* 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 ∧ 𝐵𝐽 ∧ ∀𝑥𝐽𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)) → (topGen‘𝐵) = 𝐽)

Theorem2basgen 20842 Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 2-Sep-2015.)
((𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = (topGen‘𝐶))

Theoremtgfiss 20843 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 ∧ 𝐴𝐽) → (topGen‘(fi‘𝐴)) ⊆ 𝐽)

Theoremtgdif0 20844 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)

Theorembastop1 20845* 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 ∧ 𝐵𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥𝐽𝑦(𝑦𝐵𝑥 = 𝑦)))

Theorembastop2 20846* A version of bastop1 20845 that doesn't have 𝐵𝐽 in the antecedent. (Contributed by NM, 3-Feb-2008.)
(𝐽 ∈ Top → ((topGen‘𝐵) = 𝐽 ↔ (𝐵𝐽 ∧ ∀𝑥𝐽𝑦(𝑦𝐵𝑥 = 𝑦))))

12.1.3  Examples of topologies

Theoremdistop 20847 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.)
(𝐴𝑉 → 𝒫 𝐴 ∈ Top)

Theoremtopnex 20848 The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 7010; an alternate proof uses indiscrete topologies (see indistop 20854) and the analogue of pwnex 7010 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7007). (Contributed by BJ, 2-May-2021.)
Top ∉ V

Theoremdistopon 20849 The discrete topology on a set 𝐴, with base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))

Theoremsn0topon 20850 The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.)
{∅} ∈ (TopOn‘∅)

Theoremsn0top 20851 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

Theoremindislem 20852 A lemma to eliminate some sethood hypotheses when dealing with the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
{∅, ( I ‘𝐴)} = {∅, 𝐴}

Theoremindistopon 20853 The indiscrete topology on a set 𝐴. Part of Example 2 in [Munkres] p. 77. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐴𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))

Theoremindistop 20854 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

Theoremindisuni 20855 The base set of the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
( I ‘𝐴) = {∅, 𝐴}

Theoremfctop 20856* 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.)
(𝐴𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ∈ Fin ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))

Theoremfctop2 20857* The finite complement topology on a set 𝐴. Example 3 in [Munkres] p. 77. (This version of fctop 20856 requires the Axiom of Infinity.) (Contributed by FL, 20-Aug-2006.)
(𝐴𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≺ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))

Theoremcctop 20858* 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.)
(𝐴𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))

Theoremppttop 20859* The particular point topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ∈ (TopOn‘𝐴))

Theorempptbas 20860* The particular point topology is generated by a basis consisting of pairs {𝑥, 𝑃} for each 𝑥𝐴. (Contributed by Mario Carneiro, 3-Sep-2015.)
((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} = (topGen‘ran (𝑥𝐴 ↦ {𝑥, 𝑃})))

Theoremepttop 20861* The excluded point topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ (TopOn‘𝐴))

Theoremindistpsx 20862 The indiscrete topology on a set 𝐴 expressed as a topological space, using explicit structure component references. Compare with indistps 20863 and indistps2 20864. The advantage of this version is that the actual function for the structure is evident, and df-ndx 15907 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 15910 and df-tset 16007 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 20863 instead. (New usage is discouraged.) (Contributed by FL, 19-Jul-2006.)
𝐴 ∈ V    &   𝐾 = {⟨1, 𝐴⟩, ⟨9, {∅, 𝐴}⟩}       𝐾 ∈ TopSp

Theoremindistps 20863 The indiscrete topology on a set 𝐴 expressed as a topological space, using implicit structure indices. The advantage of this version over indistpsx 20862 is that it is independent of the indices of the component definitions df-base 15910 and df-tset 16007, and if they are changed in the future, this theorem will not be affected. The advantage over indistps2 20864 is that it is easy to eliminate the hypotheses with eqid 2651 and vtoclg 3297 to result in a closed theorem. Theorems indistpsALT 20865 and indistps2ALT 20866 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

Theoremindistps2 20864 The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Compare with indistps 20863. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 20865 and indistps2ALT 20866 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

TheoremindistpsALT 20865 The indiscrete topology on a set 𝐴 expressed as a topological space. Here we show how to derive the structural version indistps 20863 from the direct component assignment version indistps2 20864. (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

Theoremindistps2ALT 20866 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 20864 from the structural version indistps 20863. (Contributed by NM, 24-Oct-2012.) (New usage is discouraged.) (Proof modification is discouraged.)
(Base‘𝐾) = 𝐴    &   (TopOpen‘𝐾) = {∅, 𝐴}       𝐾 ∈ TopSp

Theoremdistps 20867 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

Syntaxccld 20868 Extend class notation with the set of closed sets of a topology.
class Clsd

Syntaxcnt 20869 Extend class notation with interior of a subset of a topology base set.
class int

Syntaxccl 20870 Extend class notation with closure of a subset of a topology base set.
class cls

Definitiondf-cld 20871* 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 ∣ ( 𝑗𝑥) ∈ 𝑗})

Definitiondf-ntr 20872* Define a function on topologies whose value is the interior function on the subsets of the base set. See ntrval 20888. (Contributed by NM, 10-Sep-2006.)
int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 (𝑗 ∩ 𝒫 𝑥)))

Definitiondf-cls 20873* Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 20889. (Contributed by NM, 3-Oct-2006.)
cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦}))

Theoremfncld 20874 The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Clsd Fn Top

Theoremcldval 20875* 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 ∣ (𝑋𝑥) ∈ 𝐽})

Theoremntrfval 20876* 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)))

Theoremclsfval 20877* 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) ∣ 𝑥𝑦}))

Theoremcldrcl 20878 Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)

Theoremiscld 20879 The predicate "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) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))

Theoremiscld2 20880 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 ∧ 𝑆𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑋𝑆) ∈ 𝐽))

Theoremcldss 20881 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) → 𝑆𝑋)

Theoremcldss2 20882 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

Theoremcldopn 20883 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) → (𝑋𝑆) ∈ 𝐽)

Theoremisopn2 20884 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 ∧ 𝑆𝑋) → (𝑆𝐽 ↔ (𝑋𝑆) ∈ (Clsd‘𝐽)))

Theoremopncld 20885 The complement of an open set is closed. (Contributed by NM, 6-Oct-2006.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝐽) → (𝑋𝑆) ∈ (Clsd‘𝐽))

Theoremdifopn 20886 The difference of a closed set with an open set is open. (Contributed by Mario Carneiro, 6-Jan-2014.)
𝑋 = 𝐽       ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝐵) ∈ 𝐽)

Theoremtopcld 20887 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))

Theoremntrval 20888 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 ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) = (𝐽 ∩ 𝒫 𝑆))

Theoremclsval 20889* 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 ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})

Theorem0cld 20890 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))

Theoremiincld 20891* 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 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))

Theoremintcld 20892 The intersection of a set of closed sets is closed. (Contributed by NM, 5-Oct-2006.)
((𝐴 ≠ ∅ ∧ 𝐴 ⊆ (Clsd‘𝐽)) → 𝐴 ∈ (Clsd‘𝐽))

Theoremuncld 20893 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)) → (𝐴𝐵) ∈ (Clsd‘𝐽))

Theoremcldcls 20894 A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.)
(𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆)

Theoremincld 20895 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)) → (𝐴𝐵) ∈ (Clsd‘𝐽))

Theoremriincld 20896* An indexed relative intersection of closed sets is closed. (Contributed by Stefan O'Rear, 22-Feb-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 𝑥𝐴 𝐵) ∈ (Clsd‘𝐽))

Theoremiuncld 20897* A finite indexed union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))

Theoremunicld 20898 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))

Theoremclscld 20899 The closure of a subset of a topology's underlying set is closed. (Contributed by NM, 4-Oct-2006.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))

Theoremclsf 20900 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))

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