Theorem List for Intuitionistic Logic Explorer - 12201-12300 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | tpsuni 12201 |
The base set of a topological space. (Contributed by FL,
27-Jun-2014.)
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Theorem | tpstop 12202 |
The topology extractor on a topological space is a topology.
(Contributed by FL, 27-Jun-2014.)
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Theorem | tpspropd 12203 |
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 12204 |
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 12205 |
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 12206 |
Properties that determine a topological space. (Contributed by NM,
20-Oct-2012.)
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Theorem | eltpsg 12207 |
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 12208 |
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 12209 |
Syntax for the class of topological bases.
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Definition | df-bases 12210* |
Define the class of topological bases. Equivalent to definition of
basis in [Munkres] p. 78 (see isbasis2g 12212). Note that "bases" is the
plural of "basis". (Contributed by NM, 17-Jul-2006.)
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Theorem | isbasisg 12211* |
Express the predicate "the set is a basis for a topology".
(Contributed by NM, 17-Jul-2006.)
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Theorem | isbasis2g 12212* |
Express the predicate "the set is a basis for a topology".
(Contributed by NM, 17-Jul-2006.)
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Theorem | isbasis3g 12213* |
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 12214 |
Property of a basis. (Contributed by NM, 16-Jul-2006.)
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Theorem | basis2 12215* |
Property of a basis. (Contributed by NM, 17-Jul-2006.)
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Theorem | fiinbas 12216* |
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 12217* |
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 12218* |
The topology generated by a basis. See also tgval2 12220 and tgval3 12227.
(Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro,
10-Jan-2015.)
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Theorem | tgvalex 12219 |
The topology generated by a basis is a set. (Contributed by Jim
Kingdon, 4-Mar-2023.)
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Theorem | tgval2 12220* |
Definition of a topology generated by a basis in [Munkres] p. 78. Later
we show (in tgcl 12233) that is indeed a topology (on
, see unitg 12231). See also tgval 12218 and tgval3 12227. (Contributed
by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
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Theorem | eltg 12221 |
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 12222* |
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 12223* |
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 12224 |
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 12225 |
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 12226* |
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 12227* |
Alternate expression for the topology generated by a basis. Lemma 2.1
of [Munkres] p. 80. See also tgval 12218 and tgval2 12220. (Contributed by
NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.)
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Theorem | tg1 12228 |
Property of a member of a topology generated by a basis. (Contributed
by NM, 20-Jul-2006.)
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Theorem | tg2 12229* |
Property of a member of a topology generated by a basis. (Contributed
by NM, 20-Jul-2006.)
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Theorem | bastg 12230 |
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 12231 |
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 12232 |
Subset relation for generated topologies. (Contributed by NM,
7-May-2007.)
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Theorem | tgcl 12233 |
Show that a basis generates a topology. Remark in [Munkres] p. 79.
(Contributed by NM, 17-Jul-2006.)
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Theorem | tgclb 12234 |
The property tgcl 12233 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 12235 |
A basis generates a topology on .
(Contributed by Mario
Carneiro, 14-Aug-2015.)
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TopOn |
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Theorem | topbas 12236 |
A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
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Theorem | tgtop 12237 |
A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
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Theorem | eltop 12238 |
Membership in a topology, expressed without quantifiers. (Contributed
by NM, 19-Jul-2006.)
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Theorem | eltop2 12239* |
Membership in a topology. (Contributed by NM, 19-Jul-2006.)
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Theorem | eltop3 12240* |
Membership in a topology. (Contributed by NM, 19-Jul-2006.)
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Theorem | tgdom 12241 |
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 12242* |
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 12243 |
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 12244 |
Two ways to express that a basis is a topology. (Contributed by NM,
18-Jul-2006.)
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Theorem | tgtop11 12245 |
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 12246 |
is the only topology
with one element. (Contributed by FL,
18-Aug-2008.)
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Theorem | tgss3 12247 |
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 12248* |
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 12249 |
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 12250* |
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 12251 |
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 12252* |
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 12253* |
A version of bastop1 12252 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 12254 |
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 12255 |
The class of all topologies is a proper class. The proof uses
discrete topologies and pwnex 4370. (Contributed by BJ, 2-May-2021.)
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Theorem | distopon 12256 |
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 12257 |
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 12258 |
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 12259* |
The excluded point topology. (Contributed by Mario Carneiro,
3-Sep-2015.)
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TopOn |
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Theorem | distps 12260 |
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 12261 |
Extend class notation with the set of closed sets of a topology.
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Syntax | cnt 12262 |
Extend class notation with interior of a subset of a topology base set.
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Syntax | ccl 12263 |
Extend class notation with closure of a subset of a topology base set.
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Definition | df-cld 12264* |
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 12265* |
Define a function on topologies whose value is the interior function on
the subsets of the base set. See ntrval 12279. (Contributed by NM,
10-Sep-2006.)
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Definition | df-cls 12266* |
Define a function on topologies whose value is the closure function on
the subsets of the base set. See clsval 12280. (Contributed by NM,
3-Oct-2006.)
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Theorem | fncld 12267 |
The closed-set generator is a well-behaved function. (Contributed by
Stefan O'Rear, 1-Feb-2015.)
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Theorem | cldval 12268* |
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 12269* |
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 12270* |
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 12271 |
Reverse closure of the closed set operation. (Contributed by Stefan
O'Rear, 22-Feb-2015.)
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Theorem | iscld 12272 |
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 12273 |
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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Theorem | cldss 12274 |
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.)
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Theorem | cldss2 12275 |
The set of closed sets is contained in the powerset of the base.
(Contributed by Mario Carneiro, 6-Jan-2014.)
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Theorem | cldopn 12276 |
The complement of a closed set is open. (Contributed by NM,
5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.)
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Theorem | difopn 12277 |
The difference of a closed set with an open set is open. (Contributed
by Mario Carneiro, 6-Jan-2014.)
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Theorem | topcld 12278 |
The underlying set of a topology is closed. Part of Theorem 6.1(1) of
[Munkres] p. 93. (Contributed by NM,
3-Oct-2006.)
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Theorem | ntrval 12279 |
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.)
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Theorem | clsval 12280* |
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.)
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Theorem | 0cld 12281 |
The empty set is closed. Part of Theorem 6.1(1) of [Munkres] p. 93.
(Contributed by NM, 4-Oct-2006.)
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Theorem | uncld 12282 |
The union of two closed sets is closed. Equivalent to Theorem 6.1(3) of
[Munkres] p. 93. (Contributed by NM,
5-Oct-2006.)
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Theorem | cldcls 12283 |
A closed subset equals its own closure. (Contributed by NM,
15-Mar-2007.)
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Theorem | iuncld 12284* |
A finite indexed union of closed sets is closed. (Contributed by Mario
Carneiro, 19-Sep-2015.) (Revised by Jim Kingdon, 10-Mar-2023.)
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Theorem | unicld 12285 |
A finite union of closed sets is closed. (Contributed by Mario
Carneiro, 19-Sep-2015.)
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Theorem | ntropn 12286 |
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.)
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Theorem | clsss 12287 |
Subset relationship for closure. (Contributed by NM, 10-Feb-2007.)
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Theorem | ntrss 12288 |
Subset relationship for interior. (Contributed by NM, 3-Oct-2007.)
(Revised by Jim Kingdon, 11-Mar-2023.)
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Theorem | sscls 12289 |
A subset of a topology's underlying set is included in its closure.
(Contributed by NM, 22-Feb-2007.)
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Theorem | ntrss2 12290 |
A subset includes its interior. (Contributed by NM, 3-Oct-2007.)
(Revised by Mario Carneiro, 11-Nov-2013.)
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Theorem | ssntr 12291 |
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.)
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Theorem | ntrss3 12292 |
The interior of a subset of a topological space is included in the
space. (Contributed by NM, 1-Oct-2007.)
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Theorem | ntrin 12293 |
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.)
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Theorem | isopn3 12294 |
A subset is open iff it equals its own interior. (Contributed by NM,
9-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
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Theorem | ntridm 12295 |
The interior operation is idempotent. (Contributed by NM,
2-Oct-2007.)
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Theorem | clstop 12296 |
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.)
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Theorem | ntrtop 12297 |
The interior of a topology's underlying set is the entire set.
(Contributed by NM, 12-Sep-2006.)
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Theorem | clsss2 12298 |
If a subset is included in a closed set, so is the subset's closure.
(Contributed by NM, 22-Feb-2007.)
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Theorem | clsss3 12299 |
The closure of a subset of a topological space is included in the space.
(Contributed by NM, 26-Feb-2007.)
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Theorem | ntrcls0 12300 |
A subset whose closure has an empty interior also has an empty interior.
(Contributed by NM, 4-Oct-2007.)
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