Type  Label  Description 
Statement 

Theorem  cls0 12001 
The closure of the empty set. (Contributed by NM, 2Oct2007.) (Proof
shortened by Jim Kingdon, 12Mar2023.)



Theorem  ntr0 12002 
The interior of the empty set. (Contributed by NM, 2Oct2007.)



Theorem  isopn3i 12003 
An open subset equals its own interior. (Contributed by Mario Carneiro,
30Dec2016.)



Theorem  discld 12004 
The open sets of a discrete topology are closed and its closed sets are
open. (Contributed by FL, 7Jun2007.) (Revised by Mario Carneiro,
7Apr2015.)



Theorem  sn0cld 12005 
The closed sets of the topology .
(Contributed by FL,
5Jan2009.)



6.1.5 Neighborhoods


Syntax  cnei 12006 
Extend class notation with neighborhood relation for topologies.



Definition  dfnei 12007* 
Define a function on topologies whose value is a map from a subset to
its neighborhoods. (Contributed by NM, 11Feb2007.)



Theorem  neifval 12008* 
Value of the neighborhood function on the subsets of the base set of a
topology. (Contributed by NM, 11Feb2007.) (Revised by Mario
Carneiro, 11Nov2013.)



Theorem  neif 12009 
The neighborhood function is a function from the set of the subsets of
the base set of a topology. (Contributed by NM, 12Feb2007.) (Revised
by Mario Carneiro, 11Nov2013.)



Theorem  neiss2 12010 
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, 12Feb2007.)



Theorem  neival 12011* 
Value of the set of neighborhoods of a subset of the base set of a
topology. (Contributed by NM, 11Feb2007.) (Revised by Mario
Carneiro, 11Nov2013.)



Theorem  isnei 12012* 
The predicate "the class is a neighborhood of ".
(Contributed by FL, 25Sep2006.) (Revised by Mario Carneiro,
11Nov2013.)



Theorem  neiint 12013 
An intuitive definition of a neighborhood in terms of interior.
(Contributed by Szymon Jaroszewicz, 18Dec2007.) (Revised by Mario
Carneiro, 11Nov2013.)



Theorem  isneip 12014* 
The predicate "the class is a neighborhood of point ".
(Contributed by NM, 26Feb2007.)



Theorem  neii1 12015 
A neighborhood is included in the topology's base set. (Contributed by
NM, 12Feb2007.)



Theorem  neisspw 12016 
The neighborhoods of any set are subsets of the base set. (Contributed
by Stefan O'Rear, 6Aug2015.)



Theorem  neii2 12017* 
Property of a neighborhood. (Contributed by NM, 12Feb2007.)



Theorem  neiss 12018 
Any neighborhood of a set is also a neighborhood of any subset
. Similar to Proposition 1 of [BourbakiTop1] p. I.2.
(Contributed by FL, 25Sep2006.)



Theorem  ssnei 12019 
A set is included in any of its neighborhoods. Generalization to
subsets of elnei 12020. (Contributed by FL, 16Nov2006.)



Theorem  elnei 12020 
A point belongs to any of its neighborhoods. Property V_{iii} of
[BourbakiTop1] p. I.3. (Contributed
by FL, 28Sep2006.)



Theorem  0nnei 12021 
The empty set is not a neighborhood of a nonempty set. (Contributed by
FL, 18Sep2007.)



Theorem  neipsm 12022* 
A neighborhood of a set is a neighborhood of every point in the set.
Proposition 1 of [BourbakiTop1] p.
I.2. (Contributed by FL,
16Nov2006.) (Revised by Jim Kingdon, 22Mar2023.)



Theorem  opnneissb 12023 
An open set is a neighborhood of any of its subsets. (Contributed by
FL, 2Oct2006.)



Theorem  opnssneib 12024 
Any superset of an open set is a neighborhood of it. (Contributed by
NM, 14Feb2007.)



Theorem  ssnei2 12025 
Any subset of containing a neighborhood
of a set
is a neighborhood of this set. Generalization to subsets of Property
V_{i} of [BourbakiTop1] p. I.3. (Contributed by FL,
2Oct2006.)



Theorem  opnneiss 12026 
An open set is a neighborhood of any of its subsets. (Contributed by NM,
13Feb2007.)



Theorem  opnneip 12027 
An open set is a neighborhood of any of its members. (Contributed by NM,
8Mar2007.)



Theorem  tpnei 12028 
The underlying set of a topology is a neighborhood of any of its
subsets. Special case of opnneiss 12026. (Contributed by FL,
2Oct2006.)



Theorem  neiuni 12029 
The union of the neighborhoods of a set equals the topology's underlying
set. (Contributed by FL, 18Sep2007.) (Revised by Mario Carneiro,
9Apr2015.)



Theorem  topssnei 12030 
A finer topology has more neighborhoods. (Contributed by Mario
Carneiro, 9Apr2015.)



Theorem  innei 12031 
The intersection of two neighborhoods of a set is also a neighborhood of
the set. Generalization to subsets of Property V_{ii} of [BourbakiTop1]
p. I.3 for binary intersections. (Contributed by FL, 28Sep2006.)



Theorem  opnneiid 12032 
Only an open set is a neighborhood of itself. (Contributed by FL,
2Oct2006.)



Theorem  neissex 12033* 
For any neighborhood
of , there is a
neighborhood of
such that is a neighborhood of all
subsets of .
Generalization to subsets of Property V_{iv} of [BourbakiTop1] p. I.3.
(Contributed by FL, 2Oct2006.)



Theorem  0nei 12034 
The empty set is a neighborhood of itself. (Contributed by FL,
10Dec2006.)



6.1.6 Subspace topologies


Theorem  restrcl 12035 
Reverse closure for the subspace topology. (Contributed by Mario
Carneiro, 19Mar2015.) (Proof shortened by Jim Kingdon,
23Mar2023.)

↾_{t}


Theorem  restbasg 12036 
A subspace topology basis is a basis. (Contributed by Mario Carneiro,
19Mar2015.)

↾_{t} 

Theorem  tgrest 12037 
A subspace can be generated by restricted sets from a basis for the
original topology. (Contributed by Mario Carneiro, 19Mar2015.)
(Proof shortened by Mario Carneiro, 30Aug2015.)

↾_{t} ↾_{t} 

Theorem  resttop 12038 
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, 15Apr2007.) (Revised by Mario Carneiro,
1May2015.)

↾_{t} 

Theorem  resttopon 12039 
A subspace topology is a topology on the base set. (Contributed by
Mario Carneiro, 13Aug2015.)

TopOn
↾_{t} TopOn 

Theorem  restuni 12040 
The underlying set of a subspace topology. (Contributed by FL,
5Jan2009.) (Revised by Mario Carneiro, 13Aug2015.)

↾_{t} 

Theorem  stoig 12041 
The topological space built with a subspace topology. (Contributed by
FL, 5Jan2009.) (Proof shortened by Mario Carneiro, 1May2015.)

TopSet
↾_{t} 

Theorem  restco 12042 
Composition of subspaces. (Contributed by Mario Carneiro, 15Dec2013.)
(Revised by Mario Carneiro, 1May2015.)

↾_{t} ↾_{t} ↾_{t} 

Theorem  restabs 12043 
Equivalence of being a subspace of a subspace and being a subspace of the
original. (Contributed by Jeff Hankins, 11Jul2009.) (Proof shortened
by Mario Carneiro, 1May2015.)

↾_{t} ↾_{t} ↾_{t} 

Theorem  restin 12044 
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, 15Dec2013.)

↾_{t} ↾_{t} 

Theorem  restuni2 12045 
The underlying set of a subspace topology. (Contributed by Mario
Carneiro, 21Mar2015.)

↾_{t} 

Theorem  resttopon2 12046 
The underlying set of a subspace topology. (Contributed by Mario
Carneiro, 13Aug2015.)

TopOn
↾_{t} TopOn 

Theorem  rest0 12047 
The subspace topology induced by the topology on the empty set.
(Contributed by FL, 22Dec2008.) (Revised by Mario Carneiro,
1May2015.)

↾_{t} 

Theorem  restsn 12048 
The only subspace topology induced by the topology .
(Contributed by FL, 5Jan2009.) (Revised by Mario Carneiro,
15Dec2013.)

↾_{t}


Theorem  restopnb 12049 
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, 2Mar2015.)

↾_{t} 

Theorem  ssrest 12050 
If is a finer
topology than , then
the subspace topologies
induced by
maintain this relationship. (Contributed by Mario
Carneiro, 21Mar2015.) (Revised by Mario Carneiro, 1May2015.)

↾_{t} ↾_{t} 

Theorem  restopn2 12051 
If is open, then is open in iff it is an open subset
of
. (Contributed
by Mario Carneiro, 2Mar2015.)

↾_{t}


Theorem  restdis 12052 
A subspace of a discrete topology is discrete. (Contributed by Mario
Carneiro, 19Mar2015.)

↾_{t}


6.1.7 Limits and continuity in topological
spaces


Syntax  ccn 12053 
Extend class notation with the class of continuous functions between
topologies.



Syntax  ccnp 12054 
Extend class notation with the class of functions between topologies
continuous at a given point.



Syntax  clm 12055 
Extend class notation with a function on topological spaces whose value is
the convergence relation for limit sequences in the space.



Definition  dfcn 12056* 
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 12064 for the predicate
form. (Contributed by NM, 17Oct2006.)



Definition  dfcnp 12057* 
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, 17Oct2006.)



Definition  dflm 12058* 
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
realvalued function
converges to zero (in the standard topology on the reals) with this
definition. (Contributed by NM, 7Sep2006.)



Theorem  lmrcl 12059 
Reverse closure for the convergence relation. (Contributed by Mario
Carneiro, 7Sep2015.)



Theorem  lmfval 12060* 
The relation "sequence converges to point " in a metric
space. (Contributed by NM, 7Sep2006.) (Revised by Mario Carneiro,
21Aug2015.)

TopOn


Theorem  lmreltop 12061 
The topological space convergence relation is a relation. (Contributed
by Jim Kingdon, 25Mar2023.)



Theorem  cnfval 12062* 
The set of all continuous functions from topology to topology
. (Contributed
by NM, 17Oct2006.) (Revised by Mario Carneiro,
21Aug2015.)

TopOn
TopOn


Theorem  cnpfval 12063* 
The function mapping the points in a topology to the set of all
functions from
to topology
continuous at that point.
(Contributed by NM, 17Oct2006.) (Revised by Mario Carneiro,
21Aug2015.)

TopOn
TopOn


Theorem  iscn 12064* 
The predicate "the class is a continuous function from topology
to topology
". Definition of
continuous function in
[Munkres] p. 102. (Contributed by NM,
17Oct2006.) (Revised by Mario
Carneiro, 21Aug2015.)

TopOn
TopOn


Theorem  cnpval 12065* 
The set of all functions from topology to topology that are
continuous at a point . (Contributed by NM, 17Oct2006.)
(Revised by Mario Carneiro, 11Nov2013.)

TopOn
TopOn


Theorem  iscnp 12066* 
The predicate "the class is a continuous function from topology
to topology
at point ". Based on Theorem
7.2(g) of
[Munkres] p. 107. (Contributed by NM,
17Oct2006.) (Revised by Mario
Carneiro, 21Aug2015.)

TopOn
TopOn


Theorem  iscn2 12067* 
The predicate "the class is a continuous function from topology
to topology
". Definition of
continuous function in
[Munkres] p. 102. (Contributed by Mario
Carneiro, 21Aug2015.)



Theorem  cntop1 12068 
Reverse closure for a continuous function. (Contributed by Mario
Carneiro, 21Aug2015.)



Theorem  cntop2 12069 
Reverse closure for a continuous function. (Contributed by Mario
Carneiro, 21Aug2015.)



Theorem  iscnp3 12070* 
The predicate "the class is a continuous function from topology
to topology
at point ". (Contributed by
NM,
15May2007.)

TopOn
TopOn


Theorem  cnf 12071 
A continuous function is a mapping. (Contributed by FL, 8Dec2006.)
(Revised by Mario Carneiro, 21Aug2015.)



Theorem  cnf2 12072 
A continuous function is a mapping. (Contributed by Mario Carneiro,
21Aug2015.)

TopOn
TopOn 

Theorem  cnprcl2k 12073 
Reverse closure for a function continuous at a point. (Contributed by
Mario Carneiro, 21Aug2015.) (Revised by Jim Kingdon, 28Mar2023.)

TopOn


Theorem  cnpf2 12074 
A continuous function at point is a mapping. (Contributed by
Mario Carneiro, 21Aug2015.) (Revised by Jim Kingdon, 28Mar2023.)

TopOn
TopOn 

Theorem  tgcn 12075* 
The continuity predicate when the range is given by a basis for a
topology. (Contributed by Mario Carneiro, 7Feb2015.) (Revised by
Mario Carneiro, 22Aug2015.)

TopOn TopOn


Theorem  tgcnp 12076* 
The "continuous at a point" predicate when the range is given by a
basis
for a topology. (Contributed by Mario Carneiro, 3Feb2015.) (Revised
by Mario Carneiro, 22Aug2015.)

TopOn TopOn


Theorem  ssidcn 12077 
The identity function is a continuous function from one topology to
another topology on the same set iff the domain is finer than the
codomain. (Contributed by Mario Carneiro, 21Mar2015.) (Revised by
Mario Carneiro, 21Aug2015.)

TopOn
TopOn


Theorem  icnpimaex 12078* 
Property of a function continuous at a point. (Contributed by FL,
31Dec2006.) (Revised by Jim Kingdon, 28Mar2023.)

TopOn TopOn


Theorem  idcn 12079 
A restricted identity function is a continuous function. (Contributed
by FL, 27Dec2006.) (Proof shortened by Mario Carneiro,
21Mar2015.)

TopOn 

Theorem  lmbr 12080* 
Express the binary relation "sequence converges to point
" in a
topological space. Definition 1.41 of [Kreyszig] p. 25.
The condition
allows us to use objects
more general
than sequences when convenient; see the comment in dflm 12058.
(Contributed by Mario Carneiro, 14Nov2013.)

TopOn


Theorem  lmbr2 12081* 
Express the binary relation "sequence converges to point
" in a
metric space using an arbitrary upper set of integers.
(Contributed by Mario Carneiro, 14Nov2013.)

TopOn


Theorem  lmbrf 12082* 
Express the binary relation "sequence converges to point
" in a
metric space using an arbitrary upper set of integers.
This version of lmbr2 12081 presupposes that is a function.
(Contributed by Mario Carneiro, 14Nov2013.)

TopOn


Theorem  lmconst 12083 
A constant sequence converges to its value. (Contributed by NM,
8Nov2007.) (Revised by Mario Carneiro, 14Nov2013.)

TopOn


Theorem  lmcvg 12084* 
Convergence property of a converging sequence. (Contributed by Mario
Carneiro, 14Nov2013.)



Theorem  iscnp4 12085* 
The predicate "the class is a continuous function from topology
to topology
at point " in terms of
neighborhoods.
(Contributed by FL, 18Jul2011.) (Revised by Mario Carneiro,
10Sep2015.)

TopOn
TopOn


Theorem  cnpnei 12086* 
A condition for continuity at a point in terms of neighborhoods.
(Contributed by Jeff Hankins, 7Sep2009.)



Theorem  cnima 12087 
An open subset of the codomain of a continuous function has an open
preimage. (Contributed by FL, 15Dec2006.)



Theorem  cnco 12088 
The composition of two continuous functions is a continuous function.
(Contributed by FL, 8Dec2006.) (Revised by Mario Carneiro,
21Aug2015.)



Theorem  cnptopco 12089 
The composition of a function continuous at with a function
continuous at is continuous at . Proposition 2 of
[BourbakiTop1] p. I.9.
(Contributed by FL, 16Nov2006.) (Proof
shortened by Mario Carneiro, 27Dec2014.)



Theorem  cnclima 12090 
A closed subset of the codomain of a continuous function has a closed
preimage. (Contributed by NM, 15Mar2007.) (Revised by Mario Carneiro,
21Aug2015.)



Theorem  cnntri 12091 
Property of the preimage of an interior. (Contributed by Mario
Carneiro, 25Aug2015.)



Theorem  cnntr 12092* 
Continuity in terms of interior. (Contributed by Jeff Hankins,
2Oct2009.) (Proof shortened by Mario Carneiro, 25Aug2015.)

TopOn
TopOn


Theorem  cnss1 12093 
If the topology is
finer than , then
there are more
continuous functions from than from .
(Contributed by Mario
Carneiro, 19Mar2015.) (Revised by Mario Carneiro, 21Aug2015.)

TopOn


Theorem  cnss2 12094 
If the topology is
finer than , then
there are fewer
continuous functions into than into
from some other space.
(Contributed by Mario Carneiro, 19Mar2015.) (Revised by Mario
Carneiro, 21Aug2015.)

TopOn


Theorem  cncnpi 12095 
A continuous function is continuous at all points. One direction of
Theorem 7.2(g) of [Munkres] p. 107.
(Contributed by Raph Levien,
20Nov2006.) (Proof shortened by Mario Carneiro, 21Aug2015.)



Theorem  cnsscnp 12096 
The set of continuous functions is a subset of the set of continuous
functions at a point. (Contributed by Raph Levien, 21Oct2006.)
(Revised by Mario Carneiro, 21Aug2015.)



Theorem  cncnp 12097* 
A continuous function is continuous at all points. Theorem 7.2(g) of
[Munkres] p. 107. (Contributed by NM,
15May2007.) (Proof shortened
by Mario Carneiro, 21Aug2015.)

TopOn
TopOn


Theorem  cncnp2m 12098* 
A continuous function is continuous at all points. Theorem 7.2(g) of
[Munkres] p. 107. (Contributed by Raph
Levien, 20Nov2006.) (Revised
by Jim Kingdon, 30Mar2023.)



Theorem  cnnei 12099* 
Continuity in terms of neighborhoods. (Contributed by Thierry Arnoux,
3Jan2018.)



Theorem  cnconst2 12100 
A constant function is continuous. (Contributed by Mario Carneiro,
19Mar2015.)

TopOn
TopOn 