Theorem List for Intuitionistic Logic Explorer - 12101-12200 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | 0ntop 12101 |
The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)
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Theorem | topopn 12102 |
The underlying set of a topology is an open set. (Contributed by NM,
17-Jul-2006.)
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Theorem | eltopss 12103 |
A member of a topology is a subset of its underlying set. (Contributed
by NM, 12-Sep-2006.)
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7.1.1.2 Topologies on sets
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Syntax | ctopon 12104 |
Syntax for the function of topologies on sets.
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TopOn |
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Definition | df-topon 12105* |
Define the function that associates with a set the set of topologies on
it. (Contributed by Stefan O'Rear, 31-Jan-2015.)
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TopOn
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Theorem | funtopon 12106 |
The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.)
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TopOn |
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Theorem | istopon 12107 |
Property of being a topology with a given base set. (Contributed by
Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro,
13-Aug-2015.)
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TopOn |
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Theorem | topontop 12108 |
A topology on a given base set is a topology. (Contributed by Mario
Carneiro, 13-Aug-2015.)
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TopOn |
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Theorem | toponuni 12109 |
The base set of a topology on a given base set. (Contributed by Mario
Carneiro, 13-Aug-2015.)
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TopOn |
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Theorem | topontopi 12110 |
A topology on a given base set is a topology. (Contributed by Mario
Carneiro, 13-Aug-2015.)
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TopOn |
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Theorem | toponunii 12111 |
The base set of a topology on a given base set. (Contributed by Mario
Carneiro, 13-Aug-2015.)
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TopOn |
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Theorem | toptopon 12112 |
Alternative definition of in terms of TopOn. (Contributed
by Mario Carneiro, 13-Aug-2015.)
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TopOn |
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Theorem | toptopon2 12113 |
A topology is the same thing as a topology on the union of its open sets.
(Contributed by BJ, 27-Apr-2021.)
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TopOn |
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Theorem | topontopon 12114 |
A topology on a set is a topology on the union of its open sets.
(Contributed by BJ, 27-Apr-2021.)
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TopOn TopOn |
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Theorem | toponrestid 12115 |
Given a topology on a set, restricting it to that same set has no
effect. (Contributed by Jim Kingdon, 6-Jul-2022.)
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TopOn ↾t |
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Theorem | toponsspwpwg 12116 |
The set of topologies on a set is included in the double power set of
that set. (Contributed by BJ, 29-Apr-2021.) (Revised by Jim Kingdon,
16-Jan-2023.)
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TopOn |
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Theorem | dmtopon 12117 |
The domain of TopOn is . (Contributed by BJ,
29-Apr-2021.)
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TopOn |
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Theorem | fntopon 12118 |
The class TopOn is a function with domain . (Contributed by
BJ, 29-Apr-2021.)
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TopOn |
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Theorem | toponmax 12119 |
The base set of a topology is an open set. (Contributed by Mario
Carneiro, 13-Aug-2015.)
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TopOn |
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Theorem | toponss 12120 |
A member of a topology is a subset of its underlying set. (Contributed by
Mario Carneiro, 21-Aug-2015.)
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TopOn
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Theorem | toponcom 12121 |
If is a topology on
the base set of topology , then is a
topology on the base of . (Contributed by Mario Carneiro,
22-Aug-2015.)
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TopOn
TopOn |
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Theorem | toponcomb 12122 |
Biconditional form of toponcom 12121. (Contributed by BJ, 5-Dec-2021.)
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TopOn
TopOn |
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Theorem | topgele 12123 |
The topologies over the same set have the greatest element (the discrete
topology) and the least element (the indiscrete topology). (Contributed
by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.)
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TopOn |
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7.1.1.3 Topological spaces
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Syntax | ctps 12124 |
Syntax for the class of topological spaces.
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Definition | df-topsp 12125 |
Define the class of topological spaces (as extensible structures).
(Contributed by Stefan O'Rear, 13-Aug-2015.)
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TopOn |
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Theorem | istps 12126 |
Express the predicate "is a topological space." (Contributed by
Mario
Carneiro, 13-Aug-2015.)
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TopOn |
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Theorem | istps2 12127 |
Express the predicate "is a topological space." (Contributed by NM,
20-Oct-2012.)
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Theorem | tpsuni 12128 |
The base set of a topological space. (Contributed by FL,
27-Jun-2014.)
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Theorem | tpstop 12129 |
The topology extractor on a topological space is a topology.
(Contributed by FL, 27-Jun-2014.)
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Theorem | tpspropd 12130 |
A topological space depends only on the base and topology components.
(Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro,
13-Aug-2015.)
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Theorem | topontopn 12131 |
Express the predicate "is a topological space." (Contributed by
Mario
Carneiro, 13-Aug-2015.)
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TopSet TopOn |
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Theorem | tsettps 12132 |
If the topology component is already correctly truncated, then it forms
a topological space (with the topology extractor function coming out the
same as the component). (Contributed by Mario Carneiro,
13-Aug-2015.)
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TopSet TopOn |
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Theorem | istpsi 12133 |
Properties that determine a topological space. (Contributed by NM,
20-Oct-2012.)
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Theorem | eltpsg 12134 |
Properties that determine a topological space from a construction (using
no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.)
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TopSet
TopOn |
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Theorem | eltpsi 12135 |
Properties that determine a topological space from a construction (using
no explicit indices). (Contributed by NM, 20-Oct-2012.) (Revised by
Mario Carneiro, 13-Aug-2015.)
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TopSet
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7.1.2 Topological bases
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Syntax | ctb 12136 |
Syntax for the class of topological bases.
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Definition | df-bases 12137* |
Define the class of topological bases. Equivalent to definition of
basis in [Munkres] p. 78 (see isbasis2g 12139). Note that "bases" is the
plural of "basis". (Contributed by NM, 17-Jul-2006.)
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Theorem | isbasisg 12138* |
Express the predicate "the set is a basis for a topology".
(Contributed by NM, 17-Jul-2006.)
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Theorem | isbasis2g 12139* |
Express the predicate "the set is a basis for a topology".
(Contributed by NM, 17-Jul-2006.)
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Theorem | isbasis3g 12140* |
Express the predicate "the set is a basis for a topology".
Definition of basis in [Munkres] p. 78.
(Contributed by NM,
17-Jul-2006.)
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Theorem | basis1 12141 |
Property of a basis. (Contributed by NM, 16-Jul-2006.)
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Theorem | basis2 12142* |
Property of a basis. (Contributed by NM, 17-Jul-2006.)
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Theorem | fiinbas 12143* |
If a set is closed under finite intersection, then it is a basis for a
topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
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Theorem | baspartn 12144* |
A disjoint system of sets is a basis for a topology. (Contributed by
Stefan O'Rear, 22-Feb-2015.)
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Theorem | tgval 12145* |
The topology generated by a basis. See also tgval2 12147 and tgval3 12154.
(Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro,
10-Jan-2015.)
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Theorem | tgvalex 12146 |
The topology generated by a basis is a set. (Contributed by Jim
Kingdon, 4-Mar-2023.)
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Theorem | tgval2 12147* |
Definition of a topology generated by a basis in [Munkres] p. 78. Later
we show (in tgcl 12160) that is indeed a topology (on
, see unitg 12158). See also tgval 12145 and tgval3 12154. (Contributed
by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
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Theorem | eltg 12148 |
Membership in a topology generated by a basis. (Contributed by NM,
16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
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Theorem | eltg2 12149* |
Membership in a topology generated by a basis. (Contributed by NM,
15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
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Theorem | eltg2b 12150* |
Membership in a topology generated by a basis. (Contributed by Mario
Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 10-Jan-2015.)
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Theorem | eltg4i 12151 |
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.)
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Theorem | eltg3i 12152 |
The union of a set of basic open sets is in the generated topology.
(Contributed by Mario Carneiro, 30-Aug-2015.)
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Theorem | eltg3 12153* |
Membership in a topology generated by a basis. (Contributed by NM,
15-Jul-2006.) (Revised by Jim Kingdon, 4-Mar-2023.)
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Theorem | tgval3 12154* |
Alternate expression for the topology generated by a basis. Lemma 2.1
of [Munkres] p. 80. See also tgval 12145 and tgval2 12147. (Contributed by
NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.)
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Theorem | tg1 12155 |
Property of a member of a topology generated by a basis. (Contributed
by NM, 20-Jul-2006.)
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Theorem | tg2 12156* |
Property of a member of a topology generated by a basis. (Contributed
by NM, 20-Jul-2006.)
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Theorem | bastg 12157 |
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.)
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Theorem | unitg 12158 |
The topology generated by a basis is a topology on .
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 completely specifies the
basis it corresponds to. (Contributed by NM, 16-Jul-2006.) (Proof
shortened by OpenAI, 30-Mar-2020.)
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Theorem | tgss 12159 |
Subset relation for generated topologies. (Contributed by NM,
7-May-2007.)
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Theorem | tgcl 12160 |
Show that a basis generates a topology. Remark in [Munkres] p. 79.
(Contributed by NM, 17-Jul-2006.)
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Theorem | tgclb 12161 |
The property tgcl 12160 can be reversed: if the topology generated
by
is actually a topology, then must be a topological basis. This
yields an alternative definition of . (Contributed by
Mario Carneiro, 2-Sep-2015.)
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Theorem | tgtopon 12162 |
A basis generates a topology on .
(Contributed by Mario
Carneiro, 14-Aug-2015.)
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TopOn |
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Theorem | topbas 12163 |
A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
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Theorem | tgtop 12164 |
A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
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Theorem | eltop 12165 |
Membership in a topology, expressed without quantifiers. (Contributed
by NM, 19-Jul-2006.)
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Theorem | eltop2 12166* |
Membership in a topology. (Contributed by NM, 19-Jul-2006.)
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Theorem | eltop3 12167* |
Membership in a topology. (Contributed by NM, 19-Jul-2006.)
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Theorem | tgdom 12168 |
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.)
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Theorem | tgiun 12169* |
The indexed union of a set of basic open sets is in the generated
topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
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Theorem | tgidm 12170 |
The topology generator function is idempotent. (Contributed by NM,
18-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)
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Theorem | bastop 12171 |
Two ways to express that a basis is a topology. (Contributed by NM,
18-Jul-2006.)
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Theorem | tgtop11 12172 |
The topology generation function is one-to-one when applied to completed
topologies. (Contributed by NM, 18-Jul-2006.)
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Theorem | en1top 12173 |
is the only topology
with one element. (Contributed by FL,
18-Aug-2008.)
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Theorem | tgss3 12174 |
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.)
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Theorem | tgss2 12175* |
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.)
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Theorem | basgen 12176 |
Given a topology ,
show that a subset
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.)
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Theorem | basgen2 12177* |
Given a topology ,
show that a subset
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.)
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Theorem | 2basgeng 12178 |
Conditions that determine the equality of two generated topologies.
(Contributed by NM, 8-May-2007.) (Revised by Jim Kingdon,
5-Mar-2023.)
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Theorem | bastop1 12179* |
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 " " to express
" is a basis for
topology
" 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.)
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Theorem | bastop2 12180* |
A version of bastop1 12179 that doesn't have in the antecedent.
(Contributed by NM, 3-Feb-2008.)
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7.1.3 Examples of topologies
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Theorem | distop 12181 |
The discrete topology on a set . Part of Example 2 in [Munkres]
p. 77. (Contributed by FL, 17-Jul-2006.) (Revised by Mario Carneiro,
19-Mar-2015.)
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Theorem | topnex 12182 |
The class of all topologies is a proper class. The proof uses
discrete topologies and pwnex 4340. (Contributed by BJ, 2-May-2021.)
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Theorem | distopon 12183 |
The discrete topology on a set , with base set. (Contributed by
Mario Carneiro, 13-Aug-2015.)
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TopOn |
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Theorem | sn0topon 12184 |
The singleton of the empty set is a topology on the empty set.
(Contributed by Mario Carneiro, 13-Aug-2015.)
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TopOn |
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Theorem | sn0top 12185 |
The singleton of the empty set is a topology. (Contributed by Stefan
Allan, 3-Mar-2006.) (Proof shortened by Mario Carneiro,
13-Aug-2015.)
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Theorem | epttop 12186* |
The excluded point topology. (Contributed by Mario Carneiro,
3-Sep-2015.)
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TopOn |
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Theorem | distps 12187 |
The discrete topology on a set expressed as a topological space.
(Contributed by FL, 20-Aug-2006.)
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TopSet |
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7.1.4 Closure and interior
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Syntax | ccld 12188 |
Extend class notation with the set of closed sets of a topology.
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Syntax | cnt 12189 |
Extend class notation with interior of a subset of a topology base set.
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Syntax | ccl 12190 |
Extend class notation with closure of a subset of a topology base set.
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Definition | df-cld 12191* |
Define a function on topologies whose value is the set of closed sets of
the topology. (Contributed by NM, 2-Oct-2006.)
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Definition | df-ntr 12192* |
Define a function on topologies whose value is the interior function on
the subsets of the base set. See ntrval 12206. (Contributed by NM,
10-Sep-2006.)
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Definition | df-cls 12193* |
Define a function on topologies whose value is the closure function on
the subsets of the base set. See clsval 12207. (Contributed by NM,
3-Oct-2006.)
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Theorem | fncld 12194 |
The closed-set generator is a well-behaved function. (Contributed by
Stefan O'Rear, 1-Feb-2015.)
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Theorem | cldval 12195* |
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.)
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Theorem | ntrfval 12196* |
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.)
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Theorem | clsfval 12197* |
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.)
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Theorem | cldrcl 12198 |
Reverse closure of the closed set operation. (Contributed by Stefan
O'Rear, 22-Feb-2015.)
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Theorem | iscld 12199 |
The predicate "the class is a closed set". (Contributed by NM,
2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
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Theorem | iscld2 12200 |
A subset of the underlying set of a topology is closed iff its
complement is open. (Contributed by NM, 4-Oct-2006.)
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