Theorem List for Intuitionistic Logic Explorer - 12301-12400 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Syntax | cnt 12301 |
Extend class notation with interior of a subset of a topology base set.
|
![int int](_int.gif) |
|
Syntax | ccl 12302 |
Extend class notation with closure of a subset of a topology base set.
|
![cls cls](_cls.gif) |
|
Definition | df-cld 12303* |
Define a function on topologies whose value is the set of closed sets of
the topology. (Contributed by NM, 2-Oct-2006.)
|
![( (](lp.gif) ![{ {](lbrace.gif) ![~P ~P](scrp.gif) ![U. U.](bigcup.gif) ![( (](lp.gif) ![U. U.](bigcup.gif) ![x x](_x.gif) ![j j](_j.gif) ![} }](rbrace.gif) ![) )](rp.gif) |
|
Definition | df-ntr 12304* |
Define a function on topologies whose value is the interior function on
the subsets of the base set. See ntrval 12318. (Contributed by NM,
10-Sep-2006.)
|
![( (](lp.gif) ![( (](lp.gif) ![~P
~P](scrp.gif) ![U. U.](bigcup.gif) ![U. U.](bigcup.gif) ![( (](lp.gif)
![~P ~P](scrp.gif) ![x x](_x.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Definition | df-cls 12305* |
Define a function on topologies whose value is the closure function on
the subsets of the base set. See clsval 12319. (Contributed by NM,
3-Oct-2006.)
|
![( (](lp.gif) ![( (](lp.gif) ![~P
~P](scrp.gif) ![U. U.](bigcup.gif) ![|^| |^|](bigcap.gif) ![{ {](lbrace.gif) ![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![`
`](backtick.gif) ![j j](_j.gif)
![y y](_y.gif) ![} }](rbrace.gif) ![)
)](rp.gif) ![) )](rp.gif) |
|
Theorem | fncld 12306 |
The closed-set generator is a well-behaved function. (Contributed by
Stefan O'Rear, 1-Feb-2015.)
|
![Top Top](_top.gif) |
|
Theorem | cldval 12307* |
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.)
|
![U. U.](bigcup.gif) ![( (](lp.gif)
![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![{ {](lbrace.gif) ![~P ~P](scrp.gif)
![( (](lp.gif) ![x x](_x.gif) ![J J](_cj.gif) ![} }](rbrace.gif) ![) )](rp.gif) |
|
Theorem | ntrfval 12308* |
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.)
|
![U. U.](bigcup.gif) ![( (](lp.gif)
![( (](lp.gif) ![int int](_int.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![( (](lp.gif) ![~P ~P](scrp.gif) ![U. U.](bigcup.gif) ![( (](lp.gif) ![~P ~P](scrp.gif) ![x x](_x.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | clsfval 12309* |
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.)
|
![U. U.](bigcup.gif) ![( (](lp.gif)
![( (](lp.gif) ![cls cls](_cls.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![( (](lp.gif) ![~P ~P](scrp.gif) ![|^| |^|](bigcap.gif) ![{ {](lbrace.gif) ![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![`
`](backtick.gif) ![J J](_cj.gif)
![y y](_y.gif) ![} }](rbrace.gif) ![)
)](rp.gif) ![) )](rp.gif) |
|
Theorem | cldrcl 12310 |
Reverse closure of the closed set operation. (Contributed by Stefan
O'Rear, 22-Feb-2015.)
|
![( (](lp.gif) ![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![` `](backtick.gif) ![J J](_cj.gif)
![Top Top](_top.gif) ![) )](rp.gif) |
|
Theorem | iscld 12311 |
The predicate "the class is a closed set". (Contributed by NM,
2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
|
![U. U.](bigcup.gif) ![( (](lp.gif)
![( (](lp.gif)
![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![( (](lp.gif) ![( (](lp.gif) ![S S](_cs.gif)
![J J](_cj.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | iscld2 12312 |
A subset of the underlying set of a topology is closed iff its
complement is open. (Contributed by NM, 4-Oct-2006.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif)
![X X](_cx.gif) ![( (](lp.gif) ![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![( (](lp.gif) ![S S](_cs.gif)
![J J](_cj.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | cldss 12313 |
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.)
|
![U. U.](bigcup.gif) ![( (](lp.gif)
![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![X X](_cx.gif) ![) )](rp.gif) |
|
Theorem | cldss2 12314 |
The set of closed sets is contained in the powerset of the base.
(Contributed by Mario Carneiro, 6-Jan-2014.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![~P ~P](scrp.gif) ![X X](_cx.gif) |
|
Theorem | cldopn 12315 |
The complement of a closed set is open. (Contributed by NM,
5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.)
|
![U. U.](bigcup.gif) ![( (](lp.gif)
![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![( (](lp.gif) ![S S](_cs.gif) ![J J](_cj.gif) ![) )](rp.gif) |
|
Theorem | difopn 12316 |
The difference of a closed set with an open set is open. (Contributed
by Mario Carneiro, 6-Jan-2014.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif) ![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![( (](lp.gif)
![B B](_cb.gif) ![J J](_cj.gif) ![) )](rp.gif) |
|
Theorem | topcld 12317 |
The underlying set of a topology is closed. Part of Theorem 6.1(1) of
[Munkres] p. 93. (Contributed by NM,
3-Oct-2006.)
|
![U. U.](bigcup.gif) ![( (](lp.gif)
![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![)
)](rp.gif) |
|
Theorem | ntrval 12318 |
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.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif)
![X X](_cx.gif) ![( (](lp.gif) ![( (](lp.gif) ![int int](_int.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![U.
U.](bigcup.gif) ![( (](lp.gif) ![~P ~P](scrp.gif) ![S S](_cs.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | clsval 12319* |
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.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif)
![X X](_cx.gif) ![( (](lp.gif) ![( (](lp.gif) ![cls cls](_cls.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![|^|
|^|](bigcap.gif) ![{ {](lbrace.gif) ![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![` `](backtick.gif) ![J J](_cj.gif)
![x x](_x.gif) ![} }](rbrace.gif) ![)
)](rp.gif) |
|
Theorem | 0cld 12320 |
The empty set is closed. Part of Theorem 6.1(1) of [Munkres] p. 93.
(Contributed by NM, 4-Oct-2006.)
|
![( (](lp.gif) ![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | uncld 12321 |
The union of two closed sets is closed. Equivalent to Theorem 6.1(3) of
[Munkres] p. 93. (Contributed by NM,
5-Oct-2006.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![( (](lp.gif)
![B B](_cb.gif)
![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | cldcls 12322 |
A closed subset equals its own closure. (Contributed by NM,
15-Mar-2007.)
|
![( (](lp.gif) ![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![( (](lp.gif) ![( (](lp.gif) ![cls cls](_cls.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![S S](_cs.gif) ![) )](rp.gif) |
|
Theorem | iuncld 12323* |
A finite indexed union of closed sets is closed. (Contributed by Mario
Carneiro, 19-Sep-2015.) (Revised by Jim Kingdon, 10-Mar-2023.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif) ![A. A.](forall.gif) ![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif)
![U_ U_](_cupbar.gif) ![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | unicld 12324 |
A finite union of closed sets is closed. (Contributed by Mario
Carneiro, 19-Sep-2015.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif) ![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif)
![U. U.](bigcup.gif) ![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![)
)](rp.gif) |
|
Theorem | ntropn 12325 |
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.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif)
![X X](_cx.gif) ![( (](lp.gif) ![( (](lp.gif) ![int int](_int.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![J J](_cj.gif) ![) )](rp.gif) |
|
Theorem | clsss 12326 |
Subset relationship for closure. (Contributed by NM, 10-Feb-2007.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif)
![S S](_cs.gif) ![( (](lp.gif) ![( (](lp.gif) ![cls cls](_cls.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![T T](_ct.gif) ![( (](lp.gif) ![( (](lp.gif) ![cls cls](_cls.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ntrss 12327 |
Subset relationship for interior. (Contributed by NM, 3-Oct-2007.)
(Revised by Jim Kingdon, 11-Mar-2023.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif)
![S S](_cs.gif) ![( (](lp.gif) ![( (](lp.gif) ![int int](_int.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![T T](_ct.gif) ![( (](lp.gif) ![( (](lp.gif) ![int int](_int.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | sscls 12328 |
A subset of a topology's underlying set is included in its closure.
(Contributed by NM, 22-Feb-2007.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif)
![X X](_cx.gif)
![( (](lp.gif) ![( (](lp.gif) ![cls cls](_cls.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ntrss2 12329 |
A subset includes its interior. (Contributed by NM, 3-Oct-2007.)
(Revised by Mario Carneiro, 11-Nov-2013.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif)
![X X](_cx.gif) ![( (](lp.gif) ![( (](lp.gif) ![int int](_int.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![S S](_cs.gif) ![) )](rp.gif) |
|
Theorem | ssntr 12330 |
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.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif) ![( (](lp.gif)
![X X](_cx.gif) ![( (](lp.gif)
![S S](_cs.gif) ![) )](rp.gif)
![( (](lp.gif) ![(
(](lp.gif) ![int int](_int.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ntrss3 12331 |
The interior of a subset of a topological space is included in the
space. (Contributed by NM, 1-Oct-2007.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif)
![X X](_cx.gif) ![( (](lp.gif) ![( (](lp.gif) ![int int](_int.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![X X](_cx.gif) ![) )](rp.gif) |
|
Theorem | ntrin 12332 |
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.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif)
![X X](_cx.gif) ![( (](lp.gif) ![( (](lp.gif) ![int int](_int.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![int int](_int.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![( (](lp.gif) ![( (](lp.gif) ![int int](_int.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | isopn3 12333 |
A subset is open iff it equals its own interior. (Contributed by NM,
9-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif)
![X X](_cx.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![int int](_int.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![S S](_cs.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ntridm 12334 |
The interior operation is idempotent. (Contributed by NM,
2-Oct-2007.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif)
![X X](_cx.gif) ![( (](lp.gif) ![( (](lp.gif) ![int int](_int.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![( (](lp.gif) ![( (](lp.gif) ![int int](_int.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif)
![( (](lp.gif) ![(
(](lp.gif) ![int int](_int.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | clstop 12335 |
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.)
|
![U. U.](bigcup.gif) ![( (](lp.gif)
![( (](lp.gif) ![(
(](lp.gif) ![cls cls](_cls.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![X X](_cx.gif) ![X X](_cx.gif) ![) )](rp.gif) |
|
Theorem | ntrtop 12336 |
The interior of a topology's underlying set is the entire set.
(Contributed by NM, 12-Sep-2006.)
|
![U. U.](bigcup.gif) ![( (](lp.gif)
![( (](lp.gif) ![(
(](lp.gif) ![int int](_int.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![X X](_cx.gif) ![X X](_cx.gif) ![) )](rp.gif) |
|
Theorem | clsss2 12337 |
If a subset is included in a closed set, so is the subset's closure.
(Contributed by NM, 22-Feb-2007.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif) ![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![C C](_cc.gif) ![( (](lp.gif) ![( (](lp.gif) ![cls cls](_cls.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![C C](_cc.gif) ![) )](rp.gif) |
|
Theorem | clsss3 12338 |
The closure of a subset of a topological space is included in the space.
(Contributed by NM, 26-Feb-2007.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif)
![X X](_cx.gif) ![( (](lp.gif) ![( (](lp.gif) ![cls cls](_cls.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![X X](_cx.gif) ![) )](rp.gif) |
|
Theorem | ntrcls0 12339 |
A subset whose closure has an empty interior also has an empty interior.
(Contributed by NM, 4-Oct-2007.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif)
![( (](lp.gif) ![( (](lp.gif) ![int int](_int.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![( (](lp.gif) ![( (](lp.gif) ![cls cls](_cls.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif)
![(/) (/)](varnothing.gif) ![( (](lp.gif) ![( (](lp.gif) ![int int](_int.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![(/) (/)](varnothing.gif) ![) )](rp.gif) |
|
Theorem | ntreq0 12340* |
Two ways to say that a subset has an empty interior. (Contributed by
NM, 3-Oct-2007.) (Revised by Jim Kingdon, 11-Mar-2023.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif)
![X X](_cx.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![int int](_int.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif)
![A. A.](forall.gif) ![( (](lp.gif)
![(/) (/)](varnothing.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | cls0 12341 |
The closure of the empty set. (Contributed by NM, 2-Oct-2007.) (Proof
shortened by Jim Kingdon, 12-Mar-2023.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![cls cls](_cls.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![(/) (/)](varnothing.gif) ![(/) (/)](varnothing.gif) ![) )](rp.gif) |
|
Theorem | ntr0 12342 |
The interior of the empty set. (Contributed by NM, 2-Oct-2007.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![int int](_int.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![(/) (/)](varnothing.gif) ![(/) (/)](varnothing.gif) ![) )](rp.gif) |
|
Theorem | isopn3i 12343 |
An open subset equals its own interior. (Contributed by Mario Carneiro,
30-Dec-2016.)
|
![( (](lp.gif) ![( (](lp.gif) ![J J](_cj.gif) ![( (](lp.gif) ![( (](lp.gif) ![int int](_int.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![S S](_cs.gif) ![) )](rp.gif) |
|
Theorem | discld 12344 |
The open sets of a discrete topology are closed and its closed sets are
open. (Contributed by FL, 7-Jun-2007.) (Revised by Mario Carneiro,
7-Apr-2015.)
|
![( (](lp.gif) ![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![`
`](backtick.gif) ![~P ~P](scrp.gif) ![A A](_ca.gif) ![~P ~P](scrp.gif) ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | sn0cld 12345 |
The closed sets of the topology ![{
{](lbrace.gif) ![(/) (/)](varnothing.gif) .
(Contributed by FL,
5-Jan-2009.)
|
![( (](lp.gif) ![Clsd Clsd](_clsd.gif) ![`
`](backtick.gif) ![{ {](lbrace.gif) ![(/) (/)](varnothing.gif) ![} }](rbrace.gif) ![{ {](lbrace.gif) ![(/) (/)](varnothing.gif) ![} }](rbrace.gif) |
|
7.1.5 Neighborhoods
|
|
Syntax | cnei 12346 |
Extend class notation with neighborhood relation for topologies.
|
![nei nei](_nei.gif) |
|
Definition | df-nei 12347* |
Define a function on topologies whose value is a map from a subset to
its neighborhoods. (Contributed by NM, 11-Feb-2007.)
|
![( (](lp.gif) ![( (](lp.gif) ![~P
~P](scrp.gif) ![U. U.](bigcup.gif) ![{ {](lbrace.gif) ![~P ~P](scrp.gif) ![U. U.](bigcup.gif) ![E.
E.](exists.gif) ![( (](lp.gif)
![y y](_y.gif) ![) )](rp.gif) ![} }](rbrace.gif) ![)
)](rp.gif) ![) )](rp.gif) |
|
Theorem | neifval 12348* |
Value of the neighborhood function on the subsets of the base set of a
topology. (Contributed by NM, 11-Feb-2007.) (Revised by Mario
Carneiro, 11-Nov-2013.)
|
![U. U.](bigcup.gif) ![( (](lp.gif)
![( (](lp.gif) ![nei nei](_nei.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![( (](lp.gif) ![~P ~P](scrp.gif) ![{ {](lbrace.gif) ![~P ~P](scrp.gif)
![E. E.](exists.gif) ![( (](lp.gif) ![v v](_v.gif) ![)
)](rp.gif) ![} }](rbrace.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | neif 12349 |
The neighborhood function is a function from the set of the subsets of
the base set of a topology. (Contributed by NM, 12-Feb-2007.) (Revised
by Mario Carneiro, 11-Nov-2013.)
|
![U. U.](bigcup.gif) ![( (](lp.gif)
![( (](lp.gif) ![nei nei](_nei.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![~P
~P](scrp.gif) ![X X](_cx.gif) ![) )](rp.gif) |
|
Theorem | neiss2 12350 |
A set with a neighborhood is a subset of the base set of a topology.
(This theorem depends on a function's value being empty outside of its
domain, but it will make later theorems simpler to state.) (Contributed
by NM, 12-Feb-2007.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif) ![X X](_cx.gif) ![) )](rp.gif) |
|
Theorem | neival 12351* |
Value of the set of neighborhoods of a subset of the base set of a
topology. (Contributed by NM, 11-Feb-2007.) (Revised by Mario
Carneiro, 11-Nov-2013.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif)
![X X](_cx.gif) ![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![{ {](lbrace.gif) ![~P ~P](scrp.gif)
![E. E.](exists.gif) ![( (](lp.gif) ![v v](_v.gif) ![)
)](rp.gif) ![} }](rbrace.gif) ![) )](rp.gif) |
|
Theorem | isnei 12352* |
The predicate "the class is a neighborhood of ".
(Contributed by FL, 25-Sep-2006.) (Revised by Mario Carneiro,
11-Nov-2013.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif)
![X X](_cx.gif) ![( (](lp.gif) ![( (](lp.gif) ![(
(](lp.gif) ![nei nei](_nei.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![( (](lp.gif) ![E. E.](exists.gif) ![( (](lp.gif) ![N N](_cn.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | neiint 12353 |
An intuitive definition of a neighborhood in terms of interior.
(Contributed by Szymon Jaroszewicz, 18-Dec-2007.) (Revised by Mario
Carneiro, 11-Nov-2013.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif)
![X X](_cx.gif) ![( (](lp.gif)
![( (](lp.gif) ![(
(](lp.gif) ![nei nei](_nei.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif)
![( (](lp.gif) ![( (](lp.gif) ![int int](_int.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![N N](_cn.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | isneip 12354* |
The predicate "the class is a neighborhood of point ".
(Contributed by NM, 26-Feb-2007.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif) ![X X](_cx.gif) ![( (](lp.gif) ![( (](lp.gif) ![(
(](lp.gif) ![nei nei](_nei.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![{ {](lbrace.gif) ![P P](_cp.gif) ![} }](rbrace.gif) ![( (](lp.gif)
![E. E.](exists.gif) ![( (](lp.gif) ![N N](_cn.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | neii1 12355 |
A neighborhood is included in the topology's base set. (Contributed by
NM, 12-Feb-2007.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif) ![X X](_cx.gif) ![) )](rp.gif) |
|
Theorem | neisspw 12356 |
The neighborhoods of any set are subsets of the base set. (Contributed
by Stefan O'Rear, 6-Aug-2015.)
|
![U. U.](bigcup.gif) ![( (](lp.gif)
![( (](lp.gif) ![(
(](lp.gif) ![nei nei](_nei.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![~P ~P](scrp.gif) ![X X](_cx.gif) ![) )](rp.gif) |
|
Theorem | neii2 12357* |
Property of a neighborhood. (Contributed by NM, 12-Feb-2007.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif) ![E. E.](exists.gif)
![( (](lp.gif)
![N N](_cn.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | neiss 12358 |
Any neighborhood of a set is also a neighborhood of any subset
. Similar to Proposition 1 of [BourbakiTop1] p. I.2.
(Contributed by FL, 25-Sep-2006.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![S S](_cs.gif) ![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![R R](_cr.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ssnei 12359 |
A set is included in any of its neighborhoods. Generalization to
subsets of elnei 12360. (Contributed by FL, 16-Nov-2006.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif) ![N N](_cn.gif) ![) )](rp.gif) |
|
Theorem | elnei 12360 |
A point belongs to any of its neighborhoods. Property Viii of
[BourbakiTop1] p. I.3. (Contributed
by FL, 28-Sep-2006.)
|
![( (](lp.gif) ![( (](lp.gif)
![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![{ {](lbrace.gif) ![P P](_cp.gif) ![} }](rbrace.gif) ![) )](rp.gif) ![N N](_cn.gif) ![) )](rp.gif) |
|
Theorem | 0nnei 12361 |
The empty set is not a neighborhood of a nonempty set. (Contributed by
FL, 18-Sep-2007.)
|
![( (](lp.gif) ![( (](lp.gif) ![(/) (/)](varnothing.gif) ![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | neipsm 12362* |
A neighborhood of a set is a neighborhood of every point in the set.
Proposition 1 of [BourbakiTop1] p.
I.2. (Contributed by FL,
16-Nov-2006.) (Revised by Jim Kingdon, 22-Mar-2023.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif)
![E. E.](exists.gif) ![S S](_cs.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![A. A.](forall.gif)
![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![{ {](lbrace.gif) ![p p](_p.gif) ![} }](rbrace.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | opnneissb 12363 |
An open set is a neighborhood of any of its subsets. (Contributed by
FL, 2-Oct-2006.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif) ![X X](_cx.gif) ![( (](lp.gif)
![( (](lp.gif) ![(
(](lp.gif) ![nei nei](_nei.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | opnssneib 12364 |
Any superset of an open set is a neighborhood of it. (Contributed by
NM, 14-Feb-2007.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif) ![X X](_cx.gif) ![( (](lp.gif)
![( (](lp.gif) ![(
(](lp.gif) ![nei nei](_nei.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ssnei2 12365 |
Any subset of containing a neighborhood
of a set
is a neighborhood of this set. Generalization to subsets of Property
Vi of [BourbakiTop1] p. I.3. (Contributed by FL,
2-Oct-2006.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif) ![( (](lp.gif)
![X X](_cx.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | opnneiss 12366 |
An open set is a neighborhood of any of its subsets. (Contributed by NM,
13-Feb-2007.)
|
![( (](lp.gif) ![( (](lp.gif) ![N N](_cn.gif)
![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | opnneip 12367 |
An open set is a neighborhood of any of its members. (Contributed by NM,
8-Mar-2007.)
|
![( (](lp.gif) ![( (](lp.gif)
![N N](_cn.gif) ![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![{ {](lbrace.gif) ![P P](_cp.gif) ![} }](rbrace.gif) ![) )](rp.gif) ![)
)](rp.gif) |
|
Theorem | tpnei 12368 |
The underlying set of a topology is a neighborhood of any of its
subsets. Special case of opnneiss 12366. (Contributed by FL,
2-Oct-2006.)
|
![U. U.](bigcup.gif) ![( (](lp.gif)
![( (](lp.gif)
![( (](lp.gif) ![(
(](lp.gif) ![nei nei](_nei.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | neiuni 12369 |
The union of the neighborhoods of a set equals the topology's underlying
set. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro,
9-Apr-2015.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif)
![X X](_cx.gif) ![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif) ![nei nei](_nei.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | topssnei 12370 |
A finer topology has more neighborhoods. (Contributed by Mario
Carneiro, 9-Apr-2015.)
|
![U. U.](bigcup.gif) ![U. U.](bigcup.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif)
![Y Y](_cy.gif) ![K K](_ck.gif) ![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![` `](backtick.gif) ![K K](_ck.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | innei 12371 |
The intersection of two neighborhoods of a set is also a neighborhood of
the set. Generalization to subsets of Property Vii of [BourbakiTop1]
p. I.3 for binary intersections. (Contributed by FL, 28-Sep-2006.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif) ![( (](lp.gif) ![M M](_cm.gif)
![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | opnneiid 12372 |
Only an open set is a neighborhood of itself. (Contributed by FL,
2-Oct-2006.)
|
![( (](lp.gif) ![( (](lp.gif)
![( (](lp.gif) ![(
(](lp.gif) ![nei nei](_nei.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![N N](_cn.gif)
![J J](_cj.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | neissex 12373* |
For any neighborhood
of , there is a
neighborhood of
such that is a neighborhood of all
subsets of .
Generalization to subsets of Property Viv of [BourbakiTop1] p. I.3.
(Contributed by FL, 2-Oct-2006.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif) ![E. E.](exists.gif)
![( (](lp.gif) ![(
(](lp.gif) ![nei nei](_nei.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![S S](_cs.gif) ![) )](rp.gif) ![A. A.](forall.gif) ![y y](_y.gif) ![(
(](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![`
`](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![y y](_y.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | 0nei 12374 |
The empty set is a neighborhood of itself. (Contributed by FL,
10-Dec-2006.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![nei nei](_nei.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![` `](backtick.gif) ![(/) (/)](varnothing.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
7.1.6 Subspace topologies
|
|
Theorem | restrcl 12375 |
Reverse closure for the subspace topology. (Contributed by Mario
Carneiro, 19-Mar-2015.) (Proof shortened by Jim Kingdon,
23-Mar-2023.)
|
![( (](lp.gif) ![( (](lp.gif) ↾t ![A A](_ca.gif)
![( (](lp.gif) ![_V _V](rmcv.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | restbasg 12376 |
A subspace topology basis is a basis. (Contributed by Mario Carneiro,
19-Mar-2015.)
|
![( (](lp.gif) ![( (](lp.gif) ![V V](_cv.gif) ![( (](lp.gif)
↾t ![A A](_ca.gif) ![TopBases TopBases](_topbases.gif) ![) )](rp.gif) |
|
Theorem | tgrest 12377 |
A subspace can be generated by restricted sets from a basis for the
original topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
(Proof shortened by Mario Carneiro, 30-Aug-2015.)
|
![( (](lp.gif) ![( (](lp.gif) ![W W](_cw.gif) ![( (](lp.gif) ![topGen topGen](_topgen.gif) ![` `](backtick.gif) ![( (](lp.gif) ↾t ![A A](_ca.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![topGen topGen](_topgen.gif) ![` `](backtick.gif) ![B B](_cb.gif) ↾t ![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | resttop 12378 |
A subspace topology is a topology. Definition of subspace topology in
[Munkres] p. 89. is normally a subset of the base set of
.
(Contributed by FL, 15-Apr-2007.) (Revised by Mario Carneiro,
1-May-2015.)
|
![( (](lp.gif) ![( (](lp.gif) ![V V](_cv.gif) ![( (](lp.gif)
↾t ![A A](_ca.gif) ![Top Top](_top.gif) ![) )](rp.gif) |
|
Theorem | resttopon 12379 |
A subspace topology is a topology on the base set. (Contributed by
Mario Carneiro, 13-Aug-2015.)
|
![( (](lp.gif) ![( (](lp.gif) TopOn![` `](backtick.gif) ![X X](_cx.gif) ![X X](_cx.gif) ![( (](lp.gif)
↾t ![A A](_ca.gif) TopOn![` `](backtick.gif) ![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | restuni 12380 |
The underlying set of a subspace topology. (Contributed by FL,
5-Jan-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif)
![X X](_cx.gif) ![U. U.](bigcup.gif) ![( (](lp.gif) ↾t ![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | stoig 12381 |
The topological space built with a subspace topology. (Contributed by
FL, 5-Jan-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif)
![X X](_cx.gif) ![{ {](lbrace.gif) ![<. <.](langle.gif) ![( (](lp.gif) ![Base Base](_base.gif) ![` `](backtick.gif) ![ndx ndx](_ndx.gif) ![) )](rp.gif)
![A A](_ca.gif) ![>. >.](rangle.gif) ![<.
<.](langle.gif) TopSet![`
`](backtick.gif) ![ndx ndx](_ndx.gif) ![) )](rp.gif) ![( (](lp.gif)
↾t ![A A](_ca.gif) ![) )](rp.gif) ![>. >.](rangle.gif) ![TopSp TopSp](_topsp.gif) ![)
)](rp.gif) |
|
Theorem | restco 12382 |
Composition of subspaces. (Contributed by Mario Carneiro, 15-Dec-2013.)
(Revised by Mario Carneiro, 1-May-2015.)
|
![( (](lp.gif) ![( (](lp.gif) ![X X](_cx.gif) ![( (](lp.gif) ![( (](lp.gif) ↾t ![A A](_ca.gif) ↾t ![B B](_cb.gif) ![( (](lp.gif) ↾t ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | restabs 12383 |
Equivalence of being a subspace of a subspace and being a subspace of the
original. (Contributed by Jeff Hankins, 11-Jul-2009.) (Proof shortened
by Mario Carneiro, 1-May-2015.)
|
![( (](lp.gif) ![( (](lp.gif) ![W W](_cw.gif) ![( (](lp.gif) ![( (](lp.gif) ↾t ![T T](_ct.gif) ↾t ![S S](_cs.gif) ![( (](lp.gif) ↾t ![S S](_cs.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | restin 12384 |
When the subspace region is not a subset of the base of the topology,
the resulting set is the same as the subspace restricted to the base.
(Contributed by Mario Carneiro, 15-Dec-2013.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif) ![W W](_cw.gif) ![( (](lp.gif)
↾t ![A A](_ca.gif) ![( (](lp.gif) ↾t ![( (](lp.gif) ![X X](_cx.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | restuni2 12385 |
The underlying set of a subspace topology. (Contributed by Mario
Carneiro, 21-Mar-2015.)
|
![U. U.](bigcup.gif) ![( (](lp.gif) ![(
(](lp.gif) ![V V](_cv.gif) ![( (](lp.gif) ![X X](_cx.gif)
![U. U.](bigcup.gif) ![( (](lp.gif) ↾t ![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | resttopon2 12386 |
The underlying set of a subspace topology. (Contributed by Mario
Carneiro, 13-Aug-2015.)
|
![( (](lp.gif) ![( (](lp.gif) TopOn![` `](backtick.gif) ![X X](_cx.gif)
![V V](_cv.gif) ![( (](lp.gif)
↾t ![A A](_ca.gif) TopOn![` `](backtick.gif) ![( (](lp.gif) ![X X](_cx.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | rest0 12387 |
The subspace topology induced by the topology on the empty set.
(Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro,
1-May-2015.)
|
![( (](lp.gif) ![( (](lp.gif)
↾t ![(/) (/)](varnothing.gif) ![{ {](lbrace.gif) ![(/) (/)](varnothing.gif) ![}
}](rbrace.gif) ![) )](rp.gif) |
|
Theorem | restsn 12388 |
The only subspace topology induced by the topology ![{ {](lbrace.gif) ![(/) (/)](varnothing.gif) .
(Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro,
15-Dec-2013.)
|
![( (](lp.gif) ![( (](lp.gif) ![{ {](lbrace.gif) ![(/) (/)](varnothing.gif) ↾t
![A A](_ca.gif) ![{ {](lbrace.gif) ![(/) (/)](varnothing.gif) ![} }](rbrace.gif) ![) )](rp.gif) |
|
Theorem | restopnb 12389 |
If is an open subset
of the subspace base set , then any
subset of is
open iff it is open in . (Contributed by Mario
Carneiro, 2-Mar-2015.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif)
![V V](_cv.gif)
![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![( (](lp.gif)
![( (](lp.gif) ↾t ![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ssrest 12390 |
If is a finer
topology than , then
the subspace topologies
induced by
maintain this relationship. (Contributed by Mario
Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
|
![( (](lp.gif) ![( (](lp.gif) ![K K](_ck.gif) ![( (](lp.gif) ↾t ![A A](_ca.gif) ![( (](lp.gif) ↾t ![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | restopn2 12391 |
If is open, then is open in iff it is an open subset
of
. (Contributed
by Mario Carneiro, 2-Mar-2015.)
|
![( (](lp.gif) ![( (](lp.gif) ![J J](_cj.gif) ![( (](lp.gif) ![( (](lp.gif) ↾t ![A A](_ca.gif)
![( (](lp.gif) ![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | restdis 12392 |
A subspace of a discrete topology is discrete. (Contributed by Mario
Carneiro, 19-Mar-2015.)
|
![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![( (](lp.gif) ![~P ~P](scrp.gif) ↾t ![B B](_cb.gif)
![~P ~P](scrp.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
7.1.7 Limits and continuity in topological
spaces
|
|
Syntax | ccn 12393 |
Extend class notation with the class of continuous functions between
topologies.
|
![Cn Cn](_cnf.gif) |
|
Syntax | ccnp 12394 |
Extend class notation with the class of functions between topologies
continuous at a given point.
|
![CnP CnP](_cnp.gif) |
|
Syntax | clm 12395 |
Extend class notation with a function on topological spaces whose value is
the convergence relation for limit sequences in the space.
|
![~~> ~~>](rightsquigarrow.gif) ![t t](subt.gif) |
|
Definition | df-cn 12396* |
Define a function on two topologies whose value is the set of continuous
mappings from the first topology to the second. Based on definition of
continuous function in [Munkres] p. 102.
See iscn 12405 for the predicate
form. (Contributed by NM, 17-Oct-2006.)
|
![( (](lp.gif) ![Top Top](_top.gif) ![{ {](lbrace.gif) ![( (](lp.gif) ![U. U.](bigcup.gif)
![U. U.](bigcup.gif) ![j j](_j.gif) ![A. A.](forall.gif)
![( (](lp.gif) ![`' `'](_cnv.gif) ![f f](_f.gif) ![" "](backquote.gif) ![y y](_y.gif) ![j j](_j.gif) ![} }](rbrace.gif) ![) )](rp.gif) |
|
Definition | df-cnp 12397* |
Define a function on two topologies whose value is the set of continuous
mappings at a specified point in the first topology. Based on Theorem
7.2(g) of [Munkres] p. 107.
(Contributed by NM, 17-Oct-2006.)
|
![( (](lp.gif) ![Top Top](_top.gif) ![( (](lp.gif) ![U.
U.](bigcup.gif) ![{ {](lbrace.gif) ![( (](lp.gif) ![U. U.](bigcup.gif) ![U. U.](bigcup.gif) ![j j](_j.gif) ![A. A.](forall.gif)
![( (](lp.gif) ![(
(](lp.gif) ![f f](_f.gif) ![` `](backtick.gif) ![x x](_x.gif)
![E. E.](exists.gif) ![( (](lp.gif) ![( (](lp.gif) ![f f](_f.gif) ![" "](backquote.gif) ![g g](_g.gif)
![y y](_y.gif) ![) )](rp.gif) ![)
)](rp.gif) ![} }](rbrace.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Definition | df-lm 12398* |
Define a function on topologies whose value is the convergence relation
for sequences into the given topological space. Although is
typically a sequence (a function from an upperset of integers) with
values in the topological space, it need not be. Note, however, that
the limit property concerns only values at integers, so that the
real-valued function ![( (](lp.gif) ![( (](lp.gif) ![sin sin](_sin.gif) ![` `](backtick.gif) ![( (](lp.gif) ![x x](_x.gif) ![) )](rp.gif) ![) )](rp.gif)
converges to zero (in the standard topology on the reals) with this
definition. (Contributed by NM, 7-Sep-2006.)
|
![~~> ~~>](rightsquigarrow.gif)
![( (](lp.gif) ![{ {](lbrace.gif) ![<. <.](langle.gif) ![f f](_f.gif) ![x x](_x.gif) ![( (](lp.gif) ![( (](lp.gif) ![U. U.](bigcup.gif) ![CC CC](bbc.gif) ![U.
U.](bigcup.gif) ![A. A.](forall.gif)
![( (](lp.gif)
![E. E.](exists.gif) ![ZZ>= ZZ>=](_bbzge.gif) ![( (](lp.gif) ![y y](_y.gif) ![) )](rp.gif) ![: :](colon.gif) ![y y](_y.gif) ![--> -->](longrightarrow.gif) ![u u](_u.gif) ![) )](rp.gif) ![) )](rp.gif) ![} }](rbrace.gif) ![)
)](rp.gif) |
|
Theorem | lmrcl 12399 |
Reverse closure for the convergence relation. (Contributed by Mario
Carneiro, 7-Sep-2015.)
|
![( (](lp.gif) ![F F](_cf.gif) ![( (](lp.gif) ![~~> ~~>](rightsquigarrow.gif) ![t t](subt.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![) )](rp.gif) ![Top Top](_top.gif) ![) )](rp.gif) |
|
Theorem | lmfval 12400* |
The relation "sequence converges to point " in a metric
space. (Contributed by NM, 7-Sep-2006.) (Revised by Mario Carneiro,
21-Aug-2015.)
|
![( (](lp.gif) TopOn![` `](backtick.gif) ![X X](_cx.gif) ![( (](lp.gif) ![~~> ~~>](rightsquigarrow.gif) ![t t](subt.gif) ![` `](backtick.gif) ![J J](_cj.gif) ![{ {](lbrace.gif) ![<. <.](langle.gif) ![f f](_f.gif) ![x x](_x.gif) ![( (](lp.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![A. A.](forall.gif) ![( (](lp.gif)
![E. E.](exists.gif) ![ZZ>= ZZ>=](_bbzge.gif) ![( (](lp.gif)
![y y](_y.gif) ![) )](rp.gif) ![: :](colon.gif) ![y y](_y.gif) ![--> -->](longrightarrow.gif) ![u u](_u.gif) ![) )](rp.gif) ![) )](rp.gif) ![} }](rbrace.gif) ![) )](rp.gif) |