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Type | Label | Description |
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Statement | ||
Theorem | bastop 17001 | Two ways to express that a basis is a topology. (Contributed by NM, 18-Jul-2006.) |
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Theorem | tgtop11 17002 | The topology generation function is one-to-one when applied to completed topologies. (Contributed by NM, 18-Jul-2006.) |
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Theorem | 0top 17003 | The singleton of the empty set is the only topology possible for an empty underlying set. (Contributed by NM, 9-Sep-2006.) |
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Theorem | en1top 17004 |
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Theorem | en2top 17005 | If a topology has two elements, it is the indiscrete topology. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) |
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Theorem | tgss3 17006 | 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 17007* | 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 17008 |
Given a topology ![]() ![]() |
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Theorem | basgen2 17009* |
Given a topology ![]() ![]() |
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Theorem | 2basgen 17010 | Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 2-Sep-2015.) |
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Theorem | tgfiss 17011 | 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.) |
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Theorem | tgdif0 17012 | 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.) |
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Theorem | bastop1 17013* |
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 "![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | bastop2 17014* |
A version of bastop1 17013 that doesn't have ![]() ![]() ![]() |
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Theorem | distop 17015 |
The discrete topology on a set ![]() |
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Theorem | distopon 17016 |
The discrete topology on a set ![]() |
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Theorem | sn0topon 17017 | The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
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Theorem | sn0top 17018 | 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 | indislem 17019 | A lemma to eliminate some sethood hypotheses when dealing with the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.) |
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Theorem | indistopon 17020 |
The indiscrete topology on a set ![]() |
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Theorem | indistop 17021 |
The indiscrete topology on a set ![]() |
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Theorem | indisuni 17022 | The base set of the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.) |
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Theorem | fctop 17023* |
The finite complement topology on a set ![]() |
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Theorem | fctop2 17024* |
The finite complement topology on a set ![]() |
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Theorem | cctop 17025* |
The countable complement topology on a set ![]() |
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Theorem | ppttop 17026* | The particular point topology. (Contributed by Mario Carneiro, 3-Sep-2015.) |
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Theorem | pptbas 17027* |
The particular point topology is generated by a basis consisting of
pairs ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | epttop 17028* | The excluded point topology. (Contributed by Mario Carneiro, 3-Sep-2015.) |
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Theorem | indistpsx 17029 |
The indiscrete topology on a set ![]() |
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Theorem | indistps 17030 |
The indiscrete topology on a set ![]() |
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Theorem | indistps2 17031 |
The indiscrete topology on a set ![]() |
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Theorem | indistpsALT 17032 |
The indiscrete topology on a set ![]() |
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Theorem | indistps2ALT 17033 |
The indiscrete topology on a set ![]() |
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Theorem | distps 17034 |
The discrete topology on a set ![]() |
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Syntax | ccld 17035 | Extend class notation with the set of closed sets of a topology. |
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Syntax | cnt 17036 | Extend class notation with interior of a subset of a topology base set. |
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Syntax | ccl 17037 | Extend class notation with closure of a subset of a topology base set. |
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Definition | df-cld 17038* | 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 17039* | Define a function on topologies whose value is the interior function on the subsets of the base set. See ntrval 17055. (Contributed by NM, 10-Sep-2006.) |
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Definition | df-cls 17040* | Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 17056. (Contributed by NM, 3-Oct-2006.) |
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Theorem | fncld 17041 | The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
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Theorem | cldval 17042* | 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 17043* | 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 17044* | 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 17045 | Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
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Theorem | iscld 17046 |
The predicate "![]() |
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Theorem | iscld2 17047 | 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 17048 | 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 17049 | 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 17050 | 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 | isopn2 17051 | A subset of the underlying set of a topology is open iff its complement is closed. (Contributed by NM, 4-Oct-2006.) |
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Theorem | opncld 17052 | The complement of an open set is closed. (Contributed by NM, 6-Oct-2006.) |
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Theorem | difopn 17053 | 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 17054 | 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 17055 | 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 17056* | 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 17057 | 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 | iincld 17058* |
The indexed intersection of a collection ![]() ![]() ![]() ![]() |
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Theorem | intcld 17059 | The intersection of a set of closed sets is closed. (Contributed by NM, 5-Oct-2006.) |
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Theorem | uncld 17060 | 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 17061 | A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.) |
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Theorem | incld 17062 | The intersection of two closed sets is closed. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.) |
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Theorem | riincld 17063* | An indexed relative intersection of closed sets is closed. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
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Theorem | iuncld 17064* | A finite indexed union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.) |
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Theorem | unicld 17065 | A finite union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.) |
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Theorem | clscld 17066 | The closure of a subset of a topology's underlying set is closed. (Contributed by NM, 4-Oct-2006.) |
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Theorem | clsf 17067 | The closure function is a function from subsets of the base to closed sets. (Contributed by Mario Carneiro, 11-Apr-2015.) |
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Theorem | ntropn 17068 | 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 | clsval2 17069 | Express closure in terms of interior. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
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Theorem | ntrval2 17070 | Interior expressed in terms of closure. (Contributed by NM, 1-Oct-2007.) |
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Theorem | ntrdif 17071 |
An interior of a complement is the complement of the closure. This set
is also known as the exterior of ![]() |
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Theorem | clsdif 17072 | A closure of a complement is the complement of the interior. (Contributed by Jeff Hankins, 31-Aug-2009.) |
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Theorem | clsss 17073 | Subset relationship for closure. (Contributed by NM, 10-Feb-2007.) |
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Theorem | ntrss 17074 | Subset relationship for interior. (Contributed by NM, 3-Oct-2007.) |
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Theorem | sscls 17075 | 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 17076 | A subset includes its interior. (Contributed by NM, 3-Oct-2007.) (Revised by Mario Carneiro, 11-Nov-2013.) |
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Theorem | ssntr 17077 | 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 | clsss3 17078 | 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 | ntrss3 17079 | 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 17080 | 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 | cmclsopn 17081 | The complement of a closure is open. (Contributed by NM, 11-Sep-2006.) |
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Theorem | cmntrcld 17082 | The complement of an interior is closed. (Contributed by NM, 1-Oct-2007.) |
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Theorem | iscld3 17083 | A subset is closed iff it equals its own closure. (Contributed by NM, 2-Oct-2006.) |
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Theorem | iscld4 17084 | A subset is closed iff it contains its own closure. (Contributed by NM, 31-Jan-2008.) |
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Theorem | isopn3 17085 | 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 | clsidm 17086 | The closure operation is idempotent. (Contributed by NM, 2-Oct-2007.) |
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Theorem | ntridm 17087 | The interior operation is idempotent. (Contributed by NM, 2-Oct-2007.) |
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Theorem | clstop 17088 | The closure of a topology's underlying set is entire set. (Contributed by NM, 5-Oct-2007.) |
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Theorem | ntrtop 17089 | The interior of a topology's underlying set is entire set. (Contributed by NM, 12-Sep-2006.) |
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Theorem | 0ntr 17090 | A subset with an empty interior cannot cover a whole (nonempty) topology. (Contributed by NM, 12-Sep-2006.) |
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Theorem | clsss2 17091 | 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 | elcls 17092* | Membership in a closure. Theorem 6.5(a) of [Munkres] p. 95. (Contributed by NM, 22-Feb-2007.) |
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Theorem | elcls2 17093* | Membership in a closure. (Contributed by NM, 5-Mar-2007.) |
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Theorem | clsndisj 17094 | 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.) |
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Theorem | ntrcls0 17095 | 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 17096* | Two ways to say that a subset has an empty interior. (Contributed by NM, 3-Oct-2007.) (Revised by Mario Carneiro, 11-Nov-2013.) |
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Theorem | cldmre 17097 | The closed sets of a topology comprise a Moore system on the points of the topology. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
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Theorem | mrccls 17098 | Moore closure generalizes closure in a topology. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
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Theorem | cls0 17099 | The closure of the empty set. (Contributed by NM, 2-Oct-2007.) |
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Theorem | ntr0 17100 | The interior of the empty set. (Contributed by NM, 2-Oct-2007.) |
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