Type | Label | Description |
Statement |
|
Theorem | topontopi 13601 |
A topology on a given base set is a topology. (Contributed by Mario
Carneiro, 13-Aug-2015.)
|
TopOn   |
|
Theorem | toponunii 13602 |
The base set of a topology on a given base set. (Contributed by Mario
Carneiro, 13-Aug-2015.)
|
TopOn    |
|
Theorem | toptopon 13603 |
Alternative definition of in terms of TopOn. (Contributed
by Mario Carneiro, 13-Aug-2015.)
|
 
TopOn    |
|
Theorem | toptopon2 13604 |
A topology is the same thing as a topology on the union of its open sets.
(Contributed by BJ, 27-Apr-2021.)
|
 TopOn     |
|
Theorem | topontopon 13605 |
A topology on a set is a topology on the union of its open sets.
(Contributed by BJ, 27-Apr-2021.)
|
 TopOn  TopOn     |
|
Theorem | toponrestid 13606 |
Given a topology on a set, restricting it to that same set has no
effect. (Contributed by Jim Kingdon, 6-Jul-2022.)
|
TopOn   ↾t   |
|
Theorem | toponsspwpwg 13607 |
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.)
|
 TopOn      |
|
Theorem | dmtopon 13608 |
The domain of TopOn is . (Contributed by BJ,
29-Apr-2021.)
|
TopOn  |
|
Theorem | fntopon 13609 |
The class TopOn is a function with domain . (Contributed by
BJ, 29-Apr-2021.)
|
TopOn  |
|
Theorem | toponmax 13610 |
The base set of a topology is an open set. (Contributed by Mario
Carneiro, 13-Aug-2015.)
|
 TopOn    |
|
Theorem | toponss 13611 |
A member of a topology is a subset of its underlying set. (Contributed by
Mario Carneiro, 21-Aug-2015.)
|
  TopOn 

  |
|
Theorem | toponcom 13612 |
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.)
|
  TopOn   
TopOn     |
|
Theorem | toponcomb 13613 |
Biconditional form of toponcom 13612. (Contributed by BJ, 5-Dec-2021.)
|
    TopOn  
TopOn      |
|
Theorem | topgele 13614 |
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.)
|
 TopOn          |
|
8.1.1.3 Topological spaces
|
|
Syntax | ctps 13615 |
Syntax for the class of topological spaces.
|
 |
|
Definition | df-topsp 13616 |
Define the class of topological spaces (as extensible structures).
(Contributed by Stefan O'Rear, 13-Aug-2015.)
|
     TopOn        |
|
Theorem | istps 13617 |
Express the predicate "is a topological space". (Contributed by
Mario
Carneiro, 13-Aug-2015.)
|
         TopOn    |
|
Theorem | istps2 13618 |
Express the predicate "is a topological space". (Contributed by NM,
20-Oct-2012.)
|
         
    |
|
Theorem | tpsuni 13619 |
The base set of a topological space. (Contributed by FL,
27-Jun-2014.)
|
            |
|
Theorem | tpstop 13620 |
The topology extractor on a topological space is a topology.
(Contributed by FL, 27-Jun-2014.)
|
    
  |
|
Theorem | tpspropd 13621 |
A topological space depends only on the base and topology components.
(Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro,
13-Aug-2015.)
|
                    
    |
|
Theorem | topontopn 13622 |
Express the predicate "is a topological space". (Contributed by
Mario
Carneiro, 13-Aug-2015.)
|
    TopSet   TopOn        |
|
Theorem | tsettps 13623 |
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.)
|
    TopSet   TopOn    |
|
Theorem | istpsi 13624 |
Properties that determine a topological space. (Contributed by NM,
20-Oct-2012.)
|
       
  |
|
Theorem | eltpsg 13625 |
Properties that determine a topological space from a construction (using
no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.)
|
          TopSet  
  
TopOn    |
|
Theorem | eltpsi 13626 |
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.)
|
          TopSet  
 
  |
|
8.1.2 Topological bases
|
|
Syntax | ctb 13627 |
Syntax for the class of topological bases.
|
 |
|
Definition | df-bases 13628* |
Define the class of topological bases. Equivalent to definition of
basis in [Munkres] p. 78 (see isbasis2g 13630). Note that "bases" is the
plural of "basis". (Contributed by NM, 17-Jul-2006.)
|
 
           |
|
Theorem | isbasisg 13629* |
Express the predicate "the set is a basis for a topology".
(Contributed by NM, 17-Jul-2006.)
|
               |
|
Theorem | isbasis2g 13630* |
Express the predicate "the set is a basis for a topology".
(Contributed by NM, 17-Jul-2006.)
|
    
     
     |
|
Theorem | isbasis3g 13631* |
Express the predicate "the set is a basis for a topology".
Definition of basis in [Munkres] p. 78.
(Contributed by NM,
17-Jul-2006.)
|
        
 
      
      |
|
Theorem | basis1 13632 |
Property of a basis. (Contributed by NM, 16-Jul-2006.)
|
 
           |
|
Theorem | basis2 13633* |
Property of a basis. (Contributed by NM, 17-Jul-2006.)
|
        
       |
|
Theorem | fiinbas 13634* |
If a set is closed under finite intersection, then it is a basis for a
topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
|
     
   |
|
Theorem | baspartn 13635* |
A disjoint system of sets is a basis for a topology. (Contributed by
Stefan O'Rear, 22-Feb-2015.)
|
           |
|
Theorem | tgval2 13636* |
Definition of a topology generated by a basis in [Munkres] p. 78. Later
we show (in tgcl 13649) that     is indeed a topology (on
 , see unitg 13647). See also tgval 12716 and tgval3 13643. (Contributed
by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
|
      
  
      |
|
Theorem | eltg 13637 |
Membership in a topology generated by a basis. (Contributed by NM,
16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
|
     
       |
|
Theorem | eltg2 13638* |
Membership in a topology generated by a basis. (Contributed by NM,
15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
|
                |
|
Theorem | eltg2b 13639* |
Membership in a topology generated by a basis. (Contributed by Mario
Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 10-Jan-2015.)
|
       

    |
|
Theorem | eltg4i 13640 |
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.)
|
    
      |
|
Theorem | eltg3i 13641 |
The union of a set of basic open sets is in the generated topology.
(Contributed by Mario Carneiro, 30-Aug-2015.)
|
          |
|
Theorem | eltg3 13642* |
Membership in a topology generated by a basis. (Contributed by NM,
15-Jul-2006.) (Revised by Jim Kingdon, 4-Mar-2023.)
|
              |
|
Theorem | tgval3 13643* |
Alternate expression for the topology generated by a basis. Lemma 2.1
of [Munkres] p. 80. See also tgval 12716 and tgval2 13636. (Contributed by
NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.)
|
              |
|
Theorem | tg1 13644 |
Property of a member of a topology generated by a basis. (Contributed
by NM, 20-Jul-2006.)
|
        |
|
Theorem | tg2 13645* |
Property of a member of a topology generated by a basis. (Contributed
by NM, 20-Jul-2006.)
|
        
   |
|
Theorem | bastg 13646 |
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.)
|

      |
|
Theorem | unitg 13647 |
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.)
|
     
   |
|
Theorem | tgss 13648 |
Subset relation for generated topologies. (Contributed by NM,
7-May-2007.)
|
             |
|
Theorem | tgcl 13649 |
Show that a basis generates a topology. Remark in [Munkres] p. 79.
(Contributed by NM, 17-Jul-2006.)
|
       |
|
Theorem | tgclb 13650 |
The property tgcl 13649 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.)
|
    
  |
|
Theorem | tgtopon 13651 |
A basis generates a topology on  .
(Contributed by Mario
Carneiro, 14-Aug-2015.)
|
     TopOn     |
|
Theorem | topbas 13652 |
A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
|

  |
|
Theorem | tgtop 13653 |
A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
|
       |
|
Theorem | eltop 13654 |
Membership in a topology, expressed without quantifiers. (Contributed
by NM, 19-Jul-2006.)
|
 
       |
|
Theorem | eltop2 13655* |
Membership in a topology. (Contributed by NM, 19-Jul-2006.)
|
 
 

    |
|
Theorem | eltop3 13656* |
Membership in a topology. (Contributed by NM, 19-Jul-2006.)
|
 
        |
|
Theorem | tgdom 13657 |
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.)
|
        |
|
Theorem | tgiun 13658* |
The indexed union of a set of basic open sets is in the generated
topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
|
    
      |
|
Theorem | tgidm 13659 |
The topology generator function is idempotent. (Contributed by NM,
18-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)
|
        
      |
|
Theorem | bastop 13660 |
Two ways to express that a basis is a topology. (Contributed by NM,
18-Jul-2006.)
|
     
   |
|
Theorem | tgtop11 13661 |
The topology generation function is one-to-one when applied to completed
topologies. (Contributed by NM, 18-Jul-2006.)
|
          
  |
|
Theorem | en1top 13662 |
  is the only topology
with one element. (Contributed by FL,
18-Aug-2008.)
|
 
     |
|
Theorem | tgss3 13663 |
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.)
|
       
   
       |
|
Theorem | tgss2 13664* |
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.)
|
                 

 
     |
|
Theorem | basgen 13665 |
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.)
|
 
    
   
  |
|
Theorem | basgen2 13666* |
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.)
|
 
 

         |
|
Theorem | 2basgeng 13667 |
Conditions that determine the equality of two generated topologies.
(Contributed by NM, 8-May-2007.) (Revised by Jim Kingdon,
5-Mar-2023.)
|
 
               |
|
Theorem | bastop1 13668* |
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.)
|
 
     
         |
|
Theorem | bastop2 13669* |
A version of bastop1 13668 that doesn't have in the antecedent.
(Contributed by NM, 3-Feb-2008.)
|
     
    
      |
|
8.1.3 Examples of topologies
|
|
Theorem | distop 13670 |
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.)
|
    |
|
Theorem | topnex 13671 |
The class of all topologies is a proper class. The proof uses
discrete topologies and pwnex 4451. (Contributed by BJ, 2-May-2021.)
|
 |
|
Theorem | distopon 13672 |
The discrete topology on a set , with base set. (Contributed by
Mario Carneiro, 13-Aug-2015.)
|
  TopOn    |
|
Theorem | sn0topon 13673 |
The singleton of the empty set is a topology on the empty set.
(Contributed by Mario Carneiro, 13-Aug-2015.)
|
  TopOn   |
|
Theorem | sn0top 13674 |
The singleton of the empty set is a topology. (Contributed by Stefan
Allan, 3-Mar-2006.) (Proof shortened by Mario Carneiro,
13-Aug-2015.)
|
   |
|
Theorem | epttop 13675* |
The excluded point topology. (Contributed by Mario Carneiro,
3-Sep-2015.)
|
     
 
TopOn    |
|
Theorem | distps 13676 |
The discrete topology on a set expressed as a topological space.
(Contributed by FL, 20-Aug-2006.)
|
      
   TopSet       |
|
8.1.4 Closure and interior
|
|
Syntax | ccld 13677 |
Extend class notation with the set of closed sets of a topology.
|
 |
|
Syntax | cnt 13678 |
Extend class notation with interior of a subset of a topology base set.
|
 |
|
Syntax | ccl 13679 |
Extend class notation with closure of a subset of a topology base set.
|
 |
|
Definition | df-cld 13680* |
Define a function on topologies whose value is the set of closed sets of
the topology. (Contributed by NM, 2-Oct-2006.)
|
          |
|
Definition | df-ntr 13681* |
Define a function on topologies whose value is the interior function on
the subsets of the base set. See ntrval 13695. (Contributed by NM,
10-Sep-2006.)
|
     
     |
|
Definition | df-cls 13682* |
Define a function on topologies whose value is the closure function on
the subsets of the base set. See clsval 13696. (Contributed by NM,
3-Oct-2006.)
|
         
    |
|
Theorem | fncld 13683 |
The closed-set generator is a well-behaved function. (Contributed by
Stefan O'Rear, 1-Feb-2015.)
|
 |
|
Theorem | cldval 13684* |
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.)
|
 
     
     |
|
Theorem | ntrfval 13685* |
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.)
|
 
             |
|
Theorem | clsfval 13686* |
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.)
|
 
           
    |
|
Theorem | cldrcl 13687 |
Reverse closure of the closed set operation. (Contributed by Stefan
O'Rear, 22-Feb-2015.)
|
    
  |
|
Theorem | iscld 13688 |
The predicate "the class is a closed set". (Contributed by NM,
2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
|
 

      
    |
|
Theorem | iscld2 13689 |
A subset of the underlying set of a topology is closed iff its
complement is open. (Contributed by NM, 4-Oct-2006.)
|
  
       
   |
|
Theorem | cldss 13690 |
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.)
|
 
      |
|
Theorem | cldss2 13691 |
The set of closed sets is contained in the powerset of the base.
(Contributed by Mario Carneiro, 6-Jan-2014.)
|
       |
|
Theorem | cldopn 13692 |
The complement of a closed set is open. (Contributed by NM,
5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.)
|
 
        |
|
Theorem | difopn 13693 |
The difference of a closed set with an open set is open. (Contributed
by Mario Carneiro, 6-Jan-2014.)
|
        
   |
|
Theorem | topcld 13694 |
The underlying set of a topology is closed. Part of Theorem 6.1(1) of
[Munkres] p. 93. (Contributed by NM,
3-Oct-2006.)
|
 
      |
|
Theorem | ntrval 13695 |
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.)
|
  
               |
|
Theorem | clsval 13696* |
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.)
|
  
              
   |
|
Theorem | 0cld 13697 |
The empty set is closed. Part of Theorem 6.1(1) of [Munkres] p. 93.
(Contributed by NM, 4-Oct-2006.)
|
       |
|
Theorem | uncld 13698 |
The union of two closed sets is closed. Equivalent to Theorem 6.1(3) of
[Munkres] p. 93. (Contributed by NM,
5-Oct-2006.)
|
           

      |
|
Theorem | cldcls 13699 |
A closed subset equals its own closure. (Contributed by NM,
15-Mar-2007.)
|
               |
|
Theorem | iuncld 13700* |
A finite indexed union of closed sets is closed. (Contributed by Mario
Carneiro, 19-Sep-2015.) (Revised by Jim Kingdon, 10-Mar-2023.)
|
        
       |