Theorem List for Intuitionistic Logic Explorer - 11901-12000 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | tpstop 11901 |
The topology extractor on a topological space is a topology.
(Contributed by FL, 27-Jun-2014.)
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Theorem | tpspropd 11902 |
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 11903 |
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 11904 |
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 11905 |
Properties that determine a topological space. (Contributed by NM,
20-Oct-2012.)
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Theorem | eltpsg 11906 |
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 11907 |
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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6.1.2 Topological bases
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Syntax | ctb 11908 |
Syntax for the class of topological bases.
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Definition | df-bases 11909* |
Define the class of topological bases. Equivalent to definition of
basis in [Munkres] p. 78 (see isbasis2g 11911). Note that "bases" is the
plural of "basis". (Contributed by NM, 17-Jul-2006.)
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Theorem | isbasisg 11910* |
Express the predicate "the set is a basis for a topology".
(Contributed by NM, 17-Jul-2006.)
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Theorem | isbasis2g 11911* |
Express the predicate "the set is a basis for a topology".
(Contributed by NM, 17-Jul-2006.)
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Theorem | isbasis3g 11912* |
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 11913 |
Property of a basis. (Contributed by NM, 16-Jul-2006.)
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Theorem | basis2 11914* |
Property of a basis. (Contributed by NM, 17-Jul-2006.)
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Theorem | fiinbas 11915* |
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 11916* |
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 11917* |
The topology generated by a basis. See also tgval2 11919 and tgval3 11926.
(Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro,
10-Jan-2015.)
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Theorem | tgvalex 11918 |
The topology generated by a basis is a set. (Contributed by Jim
Kingdon, 4-Mar-2023.)
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Theorem | tgval2 11919* |
Definition of a topology generated by a basis in [Munkres] p. 78. Later
we show (in tgcl 11932) that     is indeed a topology (on
 , see unitg 11930). See also tgval 11917 and tgval3 11926. (Contributed
by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
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Theorem | eltg 11920 |
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 11921* |
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 11922* |
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 11923 |
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 11924 |
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 11925* |
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 11926* |
Alternate expression for the topology generated by a basis. Lemma 2.1
of [Munkres] p. 80. See also tgval 11917 and tgval2 11919. (Contributed by
NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.)
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Theorem | tg1 11927 |
Property of a member of a topology generated by a basis. (Contributed
by NM, 20-Jul-2006.)
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Theorem | tg2 11928* |
Property of a member of a topology generated by a basis. (Contributed
by NM, 20-Jul-2006.)
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Theorem | bastg 11929 |
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 11930 |
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 11931 |
Subset relation for generated topologies. (Contributed by NM,
7-May-2007.)
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Theorem | tgcl 11932 |
Show that a basis generates a topology. Remark in [Munkres] p. 79.
(Contributed by NM, 17-Jul-2006.)
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Theorem | tgclb 11933 |
The property tgcl 11932 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 11934 |
A basis generates a topology on  .
(Contributed by Mario
Carneiro, 14-Aug-2015.)
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     TopOn     |
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Theorem | topbas 11935 |
A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
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Theorem | tgtop 11936 |
A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
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Theorem | eltop 11937 |
Membership in a topology, expressed without quantifiers. (Contributed
by NM, 19-Jul-2006.)
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Theorem | eltop2 11938* |
Membership in a topology. (Contributed by NM, 19-Jul-2006.)
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Theorem | eltop3 11939* |
Membership in a topology. (Contributed by NM, 19-Jul-2006.)
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Theorem | tgdom 11940 |
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 11941* |
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 11942 |
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 11943 |
Two ways to express that a basis is a topology. (Contributed by NM,
18-Jul-2006.)
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Theorem | tgtop11 11944 |
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 11945 |
  is the only topology
with one element. (Contributed by FL,
18-Aug-2008.)
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Theorem | tgss3 11946 |
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 11947* |
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 11948 |
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 11949* |
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 11950 |
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 11951* |
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 11952* |
A version of bastop1 11951 that doesn't have in the antecedent.
(Contributed by NM, 3-Feb-2008.)
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6.1.3 Examples of topologies
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Theorem | distop 11953 |
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 11954 |
The class of all topologies is a proper class. The proof uses
discrete topologies and pwnex 4299. (Contributed by BJ, 2-May-2021.)
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Theorem | distopon 11955 |
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 11956 |
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 11957 |
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 11958* |
The excluded point topology. (Contributed by Mario Carneiro,
3-Sep-2015.)
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TopOn    |
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Theorem | distps 11959 |
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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6.1.4 Closure and interior
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Syntax | ccld 11960 |
Extend class notation with the set of closed sets of a topology.
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Syntax | cnt 11961 |
Extend class notation with interior of a subset of a topology base set.
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Syntax | ccl 11962 |
Extend class notation with closure of a subset of a topology base set.
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Definition | df-cld 11963* |
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 11964* |
Define a function on topologies whose value is the interior function on
the subsets of the base set. See ntrval 11978. (Contributed by NM,
10-Sep-2006.)
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Definition | df-cls 11965* |
Define a function on topologies whose value is the closure function on
the subsets of the base set. See clsval 11979. (Contributed by NM,
3-Oct-2006.)
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Theorem | fncld 11966 |
The closed-set generator is a well-behaved function. (Contributed by
Stefan O'Rear, 1-Feb-2015.)
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Theorem | cldval 11967* |
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 11968* |
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 11969* |
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 11970 |
Reverse closure of the closed set operation. (Contributed by Stefan
O'Rear, 22-Feb-2015.)
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Theorem | iscld 11971 |
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 11972 |
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 11973 |
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 11974 |
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 11975 |
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 11976 |
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 11977 |
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 11978 |
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 11979* |
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 11980 |
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 11981 |
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 11982 |
A closed subset equals its own closure. (Contributed by NM,
15-Mar-2007.)
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Theorem | iuncld 11983* |
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 11984 |
A finite union of closed sets is closed. (Contributed by Mario
Carneiro, 19-Sep-2015.)
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Theorem | ntropn 11985 |
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 11986 |
Subset relationship for closure. (Contributed by NM, 10-Feb-2007.)
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Theorem | ntrss 11987 |
Subset relationship for interior. (Contributed by NM, 3-Oct-2007.)
(Revised by Jim Kingdon, 11-Mar-2023.)
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Theorem | sscls 11988 |
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 11989 |
A subset includes its interior. (Contributed by NM, 3-Oct-2007.)
(Revised by Mario Carneiro, 11-Nov-2013.)
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Theorem | ssntr 11990 |
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 11991 |
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 11992 |
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 11993 |
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 11994 |
The interior operation is idempotent. (Contributed by NM,
2-Oct-2007.)
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Theorem | clstop 11995 |
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 11996 |
The interior of a topology's underlying set is the entire set.
(Contributed by NM, 12-Sep-2006.)
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Theorem | clsss2 11997 |
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 11998 |
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 11999 |
A subset whose closure has an empty interior also has an empty interior.
(Contributed by NM, 4-Oct-2007.)
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Theorem | ntreq0 12000* |
Two ways to say that a subset has an empty interior. (Contributed by
NM, 3-Oct-2007.) (Revised by Jim Kingdon, 11-Mar-2023.)
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