Appendix:Glossary of topology

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

English Wikipedia has an article on:
Wikipedia

All spaces in this glossary are assumed to be topological spaces unless stated otherwise.

Contents: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

A edit

accessible
See  .
accumulation point
See limit point.
Alexandrov topology
A space   has the Alexandrov topology (or is finitely generated) if arbitrary intersections of open sets in   are open, or equivalently, if arbitrary unions of closed sets are closed.
almost discrete
A space is almost discrete if every open set is closed (hence clopen). The almost discrete spaces are precisely the finitely generated zero-dimensional spaces.
approach space
An approach space is a generalization of metric space based on point-to-set distances, instead of point-to-point.

B edit

Baire space
This has two distinct common meanings:
  1. A space is a Baire space if the intersection of any countable collection of dense open sets is dense; see Baire space.
  2. Baire space is the set of all functions from the natural numbers to the natural numbers, with the topology of pointwise convergence; see Baire space (set theory).
base
A collection   of open sets is a base (or basis) for a topology   if every open set in   is a union of sets in  . The topology   is the smallest topology on   containing   and is said to be generated by .
basis
See base.
Borel algebra
The Borel algebra on a topological space   is the smallest  -algebra containing all the open sets. It is obtained by taking intersection of all  -algebras on   containing  .
Borel set
A Borel set is an element of a Borel algebra.
boundary
The boundary (or frontier) of a set is the set's closure minus its interior. Equivalently, the boundary of a set is the intersection of its closure with the closure of its complement. Boundary of a set   is denoted by  
bounded
A set in a metric space is bounded if it has finite diameter. Equivalently, a set is bounded if it is contained in some open ball of finite radius. A function taking values in a metric space is bounded if its image is a bounded set.

C edit

category of topological spaces
The category Top has topological spaces as objects and continuous maps as morphisms.
cauchy sequence
A sequence   in a metric space   is a Cauchy sequence if, for every positive real number  , there is an integer   such that for all integers  , we have  .
clopen set
A set is clopen if it is both open and closed.
closed ball
If   is a metric space, a closed ball is a set of the form  , where   is in   and   is a positive real number, the radius of the ball. A closed ball of radius   is a closed  -ball. Every closed ball is a closed set in the topology induced on   by  . Note that the closed ball   might not be equal to the closure of the open ball  .
closed set
A set is closed if its complement is a member of the topology.
closed function
A function from one space to another is closed if the image of every closed set is closed.
closure
The closure of a set is the smallest closed set containing the original set. It is equal to the intersection of all closed sets which contain it. An element of the closure of a set   is a point of closure of  .
closure operator
See Kuratowski closure axioms.
coarser topology
If   is a set, and if   and   are topologies on  , then   is coarser (or smaller, weaker) than   if   is contained in  . Beware, some authors, especially analysts, use the term stronger.
comeagre
A subset   of a space   is comeagre (comeager) if its complement   is meagre. Also called residual.
compact, compact space
A space is compact if every open cover has a finite subcover. Every compact space is Lindelöf and paracompact. Therefore, every compact Hausdorff space is normal. See also quasicompact.
compact-open topology
The compact-open topology on the set   of all continuous maps between two spaces   and   is defined as follows: given a compact subset   of   and an open subset   of  , let   denote the set of all maps   in   such that   is contained in  . Then the collection of all such   is a subbase for the compact-open topology.
complete, complete space
A metric space is complete if every Cauchy sequence converges.
completely metrizable/completely metrisable
See complete space.
completely normal
A space is completely normal if any two separated sets have disjoint neighbourhoods.
completely normal Hausdorff
A completely normal Hausdorff space (or   space) is a completely normal   space. (A completely normal space is Hausdorff if and only if it is  , so the terminology is consistent.) Every completely normal Hausdorff space is normal Hausdorff.
completely regular
A space is completely regular if, whenever   is a closed set and   is a point not in  , then   and   are functionally separated.
completely  
See Tychonoff.
component
See connected component/Path-connected component.
connected
A space is connected if it is not the union of a pair of disjoint nonempty open sets. Equivalently, a space is connected if the only clopen sets are the whole space and the empty set.
connected component
A connected component of a space is a maximal nonempty connected subspace. Each connected component is closed, and the set of connected components of a space is a partition of that space.
continuous
A function from one space to another is continuous if the preimage of every open set is open.
contractible
A space   is contractible if the identity map on   is homotopic to a constant map. Every contractible space is simply connected.
coproduct topology
If   is a collection of spaces and   is the (set-theoretic) disjoint union of  , then the coproduct topology (or disjoint union topology, topological sum of the   on   is the finest topology for which all the injection maps are continuous.
countably compact
A space is countably compact if every countable open cover has a finite subcover. Every countably compact space is pseudocompact and weakly countably compact.
countably locally finite
A collection of subsets of a space   is countably locally finite (or  -locally finite) if it is the union of a countable collection of locally finite collections of subsets of  .
cover
A collection of subsets of a space is a cover (or covering) of that space if the union of the collection is the whole space.
covering
See cover.
cut point
If   is a connected space with more than one point, then a point   of   is a cut point if the subspace   is disconnected.

D edit

dense set
A set is dense if it has nonempty intersection with every nonempty open set. Equivalently, a set is dense if its closure is the whole space.
derived set
If   is a space and   is a subset of  , the derived set of   in   is the set of limit points of   in  .
diameter
If   is a metric space and   is a subset of  , the diameter of   is the supremum of the distances  , where   and   range over  .
discrete metric
The discrete metric on a set   is the function   ; R such that for all  ,   in  ,   and   if  . The discrete metric induces the discrete topology on X.
discrete space
A space  ' is discrete if every subset of   is open. We say that   carries the discrete topology.
discrete topology
See discrete space.
disjoint union topology
See coproduct topology.
dispersion point
If   is a connected space with more than one point, then a point   of   is a dispersion point if the subspace   is hereditarily disconnected (its only connected components are the one-point sets).
distance
See metric space.
dunce hat

E edit

entourage
See uniform space.
exterior
The exterior of a set is the interior of its complement.

F edit

  set
An   set is a countable union of closed sets.
filter
A filter on a space   is a nonempty family   of subsets of   such that the following conditions hold:
  1. The empty set is not in  .
  2. The intersection of any finite number of elements of   is again in  .
  3. If   is in   and if   contains  , then   is in  .
finer topology
If   is a set, and if   and   are topologies on  , then   is finer (or larger, stronger) than   if   contains  . Beware, some authors, especially analysts, use the term weaker.
finitely generated
See Alexandrov topology.
first category
See meagre.
first-countable
A space is first-countable if every point has a countable local base.
Fréchet
See  .
frontier
See boundary.
full set
A compact subset   of the complex plane is called full if its complement is connected. For example, the closed unit disk is full, while the unit circle is not.
functionally separated
Two sets   and   in a space   are functionally separated if there is a continuous map   such that   and  .

G edit

  set
A   set is a countable intersection of open sets.

H edit

Hausdorff
A Hausdorff space (or   space) is one in which every two distinct points have disjoint neighbourhoods. Every Hausdorff space is  .
hereditary
A property of spaces is said to be hereditary if whenever a space has that property, then so does every subspace of it. For example, second-countability is a hereditary property.
homeomorphism
If   and   are spaces, a homeomorphism from   to   is a bijective function   such that   and   are continuous. The spaces   and   are then said to be homeomorphic. From the standpoint of topology, homeomorphic spaces are identical.
homogeneous
A space   is homogeneous if, for every   and   in  , there is a homeomorphism  ;   such that  . Intuitively, the space looks the same at every point. Every topological group is homogeneous.
homotopic, homotopic maps
Two continuous maps   are homotopic (in  ) if there is a continuous map   such that   and   for all   in  . Here,   is given the product topology. The function   is called a homotopy (in  ) between   and  .
homotopy
See homotopic maps.
hyper-connected
A space is hyper-connected if no two non-empty open sets are disjoint. Every hyper-connected space is connected.

I edit

identification map
See quotient map.
identification space
See quotient space.
indiscrete space
See trivial topology.
Infinite-dimensional topology
See Hilbert manifods and Q-manifolds, i.e. (generalized) manifolds modelled on the Hilbert space and on the Hilbert cube respectively.
interior
The interior of a set is the largest open set contained in the original set. It is equal to the union of all open sets contained in it. An element of the interior of a set   is an interior point of  .
interior point
See interior.
isolated point
A point   is an isolated point if the singleton   is open. More generally, if   is a subset of a space  , and if   is a point of  , then   is an isolated point of   if   is open in the subspace topology on  .
isometric isomorphism
If M1 and M2 are metric spaces, an isometric isomorphism from M1 to M2 is a bijective isometry f : M1  →  M2. The metric spaces are then said to be isometrically isomorphic. From the standpoint of metric space theory, isometrically isomorphic spaces are identical.
isometry
If (M1, d1) and (M2, d2) are metric spaces, an isometry from M1 to M2 is a function f : M1  →  M2 such that d2(f(x), f(y)) = d1(x, y) for all x, y in M1. Every isometry is injective, although not every isometry is surjective.

K edit

Kolmogorov axiom
See  .
Kuratowski closure axioms
The Kuratowski closure axioms is a set of axioms satisfied by the function which takes each subset of   to its closure:
  1. isotonicity: Every set is contained in its closure.
  2. idempotence: The closure of the closure of a set is equal to the closure of that set.
  3. Preservation of binary unions: The closure of the union of two sets is the union of their closures.
  4. Preservation of nullary unions: The closure of the empty set is empty.
If   is a function from the power set of   to itself, then   is a closure operator if it satisfies the Kuratowski closure axioms. The Kuratowski closure axioms can then be used to define a topology on   by declaring the closed sets to be the fixed points of this operator, i.e. a set   is closed if and only if  .

L edit

larger topology
See finer topology.
limit point
A point   in a space   is a limit point of a subset   if every open set containing   also contains a point of   other than   itself. This is equivalent to requiring that every neighbourhood of   contains a point of   other than   itself.
limit point compact
See weakly countably compact.
Lindelöf
A space is Lindelöf if every open cover has a countable subcover.
local base
A set   of neighbourhoods of a point   of a space   is a local base (or local basis, neighbourhood base, neighbourhood basis) at   if every neighbourhood of   contains some member of  .
local basis
See local base.
locally closed subset
A subset of a topological space that is the intersection of an open and a closed subset. Equivalently, it is a relatively open subset of its closure.
locally compact
A space is locally compact if every point has a local base consisting of compact neighbourhoods. Every locally compact Hausdorff space is Tychonoff.
locally connected
A space is locally connected if every point has a local base consisting of connected neighbourhoods.
locally finite
A collection of subsets of a space is locally finite if every point has a neighbourhood which has nonempty intersection with only finitely many of the subsets. See also countably locally finite, point finite.
locally metrizable/Locally metrisable
A space is locally metrizable if every point has a metrizable neighbourhood.
locally path-connected
A space is locally path-connected if every point has a local base consisting of path-connected neighbourhoods. A locally path-connected space is connected if and only if it is path-connected.
locally simply connected
A space is locally simply connected if every point has a local base consisting of simply connected neighbourhoods.
loop
If   is a point in a space  , a loop at   in   (or a loop in  , with basepoint  ) is a path   in  , such that  . Equivalently, a loop in   is a continuous map from the unit circle  , into  .

M edit

meagre
If   is a space and   is a subset of  , then   is meagre in   (or of first category in  ) if it is the countable union of nowhere dense sets. If   is not meagre in  ,   is of second category in  .
metric
See metric space.
metric invariant
A metric invariant is a property which is preserved under isometric isomorphism.
metric map
If   and   are metric spaces with metrics   and   respectively, then a metric map is a function   from   to  , such that for any points   and   in  ,  . A metric map is strictly metric if the above inequality is strict for all   and   in  .
metric space
A metric space   is a set   equipped with a function     satisfying the following axioms for all  ,   and   in  :
  1.  
  2.  
  3.   — identity of indiscernibles
  4.   — symmetry
  5.  triangle inequality
The function   is a metric on  , and   is the distance between   and  . The collection of all open balls of M is a base for a topology on  ; this is the topology on   induced by  . Every metric space is Hausdorff and paracompact (and hence normal and Tychonoff). Every metric space is first-countable.
metrizable/Metrisable
A space is metrizable if it is homeomorphic to a metric space. Every metrizable space is Hausdorff and paracompact (and hence normal and Tychonoff). Every metrizable space is first-countable.
monolith
Every non-empty ultra-connected compact space   has a largest proper open subset; this subset is called a monolith.

N edit

neighbourhood/neighborhood
A neighbourhood of a point   is a set containing an open set which in turn contains the point  . More generally, a neighbourhood of a set   is a set containing an open set which in turn contains the set  . A neighbourhood of a point   is thus a neighbourhood of the singleton set  . (Note that under this definition, the neighbourhood itself need not be open. Many authors require that neighbourhoods be open; be careful to note conventions.)
neighbourhood base/basis
See local base.
neighbourhood system
A neighbourhood system at a point   in a space is the collection of all neighbourhoods of  .
net
A net in a space   is a map from a directed set   to  . A net from   to   is usually denoted ( ), where   is an index variable ranging over  . Every sequence is a net, taking   to be the directed set of natural numbers with the usual ordering.
normal
A space is normal if any two disjoint closed sets have disjoint neighbourhoods. Every normal space admits a partition of unity.
Normal Hausdorff
A normal Hausdorff space (or   space) is a normal   space. (A normal space is Hausdorff if and only if it is  , so the terminology is consistent.) Every normal Hausdorff space is Tychonoff.
nowhere dense
A nowhere dense set is a set whose closure has empty interior.

O edit

open cover
An open cover is a cover consisting of open sets.
open ball
If   is a metric space, an open ball is a set of the form  , where   is in   and   is a positive real number, the radius of the ball. An open ball of radius   is an open  -ball. Every open ball is an open set in the topology on   induced by  .
open condition
See open property.
open set
An open set is a member of the topology.
open function
A function from one space to another is open if the image of every open set is open.
open property
A property of points in a topological space is said to be "open" if those points which possess it form an open set. Such conditions often take a common form, and that form can be said to be an open condition; for example, in metric spaces, one defines an open ball as above, and says that "strict inequality is an open condition".

P edit

paracompact
A space is paracompact if every open cover has a locally finite open refinement. Paracompact Hausdorff spaces are normal.
partition of unity
A partition of unity of a space   is a set of continuous functions from   to   such that any point has a neighbourhood where all but a finite number of the functions are identically zero, and the sum of all the functions on the entire space is identically  .
path
A path in a space   is a continuous map   from the closed unit interval [0, 1] into  . The point   is the initial point of  ; the point   is the terminal point of  .
path-connected
A space   is path-connected if, for every two points  ,   in  , there is a path   from   to  , i.e., a path with initial point   and terminal point  . Every path-connected space is connected.
path-connected component
A path-connected component of a space is a maximal nonempty path-connected subspace. The set of path-connected components of a space is a partition of that space, which is finer than the partition into connected components. The set of path-connected components of a space X is denoted  .
point
A point is an element of a topological space. More generally, a point is an element of any set with an underlying topological structure; e.g. an element of a metric space or a topological group is also a "point".
point of closure
See closure.
polish
A space is Polish if it is separable and topologically complete, i.e. if it is homeomorphic to a separable and complete metric space.
pre-compact
See relatively compact.
product topology
If   is a collection of spaces and   is the (set-theoretic) product of  , then the product topology on   is the coarsest topology for which all the projection maps are continuous.
proper function/mapping
A continuous function f from a space   to a space   is proper if   is a compact set in   for any compact subspace   of  .
proximity space
A proximity space   is a set   equipped with a binary relation   between subsets of   satisfying the following properties:
For all subsets  ,   and   of  ,
  1.   implies  
  2.   implies   is non-empty
  3. If   and   have non-empty intersection, then  
  4.   iff (  or  )
  5. If, for all subsets E of X, we have (  or  ), then we must have  
pseudocompact
A space is pseudocompact if every real-valued continuous function on the space is bounded.
pseudometric
See pseudometric space.
pseudometric space
A pseudometric space   is a set   equipped with a function     satisfying all the conditions of a metric space, except possibly the identity of indiscernibles. That is, points in a pseudometric space may be "infinitely close" without being identical. The function   is a pseudometric on  . Every metric is a pseudometric.
punctured neighbourhood/punctured neighborhood
A punctured neighbourhood of a point   is a neighbourhood of  , minus  . For instance, the interval   is a neighbourhood of   in the real line, so the set   is a punctured neighbourhood of  .

Q edit

quasicompact
See compact. Some authors define "compact" to include the Hausdorff separation axiom, and they use the term quasicompact to mean what we call in this glossary simply "compact" (without the Hausdorff axiom). This convention is most commonly found in French, and branches of mathematics heavily influenced by the French.
quotient map
If   and   are spaces, and if   is a surjection from   to  , then   is a quotient map (or identification map) if, for every subset   of  ,   is open in   if and only if   is open in  . In other words,   has the  -strong topology. Equivalently,   is a quotient map if and only if it is the transfinite composition of maps  , where   is a subset. Note that this doesn't imply that f is an open function.
quotient space
If   is a space,   is a set, and   is any surjective function, then the quotient topology on   induced by   is the finest topology for which   is continuous. The space   is a quotient space or identification space. By definition,   is a quotient map. The most common example of this is to consider an equivalence relation on  , with   the set of equivalence classes and   the natural projection map. This construction is dual to the construction of the subspace topology.

R edit

refinement
A cover   is a refinement of a cover   if every member of   is a subset of some member of  .
regular
A space is regular if, whenever   is a closed set and   is a point not in  , then   and   have disjoint neighbourhoods.
regular Hausdorff
A space is regular Hausdorff (or  ) if it is a regular   space. (A regular space is Hausdorff if and only if it is  , so the terminology is consistent.)
regular open
An open subset   of a space   is regular open if it equals the interior of its closure. An example of a non-regular open set is the set   with its normal topology, since 1 is in the interior of the closure of U, but not in U. The regular open subsets of a space form a complete Boolean algebra.
relatively compact
A subset   of a space   is relatively compact in   if the closure of   in   is compact.
residual
If   is a space and   is a subset of  , then   is residual in   if the complement of   is meagre in  . Also called comeagre or comeager.

S edit

Second category
See meagre.
second-countable
A space is second-countable if it has a countable base for its topology. Every second-countable space is first-countable, separable, and Lindelöf.
semilocally simply connected
A space   is semilocally simply connected if, for every point   in  , there is a neighbourhood   of   such that every loop at   in   is homotopic in   to the constant loop  . Every simply connected space and every locally simply connected space is semilocally simply connected. (Compare with locally simply connected; here, the homotopy is allowed to live in  , whereas in the definition of locally simply connected, the homotopy must live in  .)
separable
A space is separable if it has a countable dense subset.
separated
Two sets   and   are separated if each is disjoint from the other's closure.
sequentially compact
A space is sequentially compact if every sequence has a convergent subsequence. Every sequentially compact space is countably compact, and every first-countable, countably compact space is sequentially compact.
short map
See metric map
simply connected
A space is simply connected if it is path-connected and every loop is homotopic to a constant map.
smaller topology
See coarser topology.
 -Strong topology
Let   be a map of topological spaces. We say that   has the  -strong topology if, for every subset  , one has that   is open in   if and only if   is open in  
stronger topology
See finer topology. Beware, some authors, especially analysts, use the term weaker topology.
subbase
A collection of open sets is a subbase (or subbasis) for a topology if every non-empty proper open set in the topology is a union of finite intersections of sets in the subbase. If   is any collection of subsets of a set  , the topology on   generated by   is the smallest topology containing  ; this topology consists of the empty set,   and all unions of finite intersections of elements of  .
subbasis
See subbase.
subcover
A cover   is a subcover (or subcovering) of a cover   if every member of   is a member of  .
subcovering
See subcover.
subspace
If T is a topology on a space  , and if   is a subset of  , then the subspace topology on   induced by   consists of all intersections of open sets in   with  . This construction is dual to the construction of the quotient topology.

T edit

 
A space is   (or Kolmogorov) if for every pair of distinct points x and y in the space, either there is an open set containing x but not y, or there is an open set containing y but not x.
 
A space is   (or Fréchet or accessible) if for every pair of distinct points x and y in the space, there is an open set containing x but not y. (Compare with  ; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is   if all its singletons are closed. Every   space is  .
 
See Hausdorff space.
 
See regular Hausdorff.
 
See Tychonoff space.
 
See normal Hausdorff.
 
See completely normal Hausdorff.
top
See category of topological spaces.
topological invariant
A topological invariant is a property which is preserved under homeomorphism. For example, compactness and connectedness are topological properties, whereas boundedness and completeness are not. Algebraic topology is the study of topologically invariant abstract algebra constructions on topological spaces.
topological space
A topological space   is a set   equipped with a collection   of subsets of   satisfying the following axioms:
  1. The empty set and   are in  .
  2. The union of any collection of sets in   is also in  .
  3. The intersection of any pair of sets in   is also in  .
The collection   is a topology on  .
topological sum
See coproduct topology.
topologically complete
A space is topologically complete if it is homeomorphic to a complete metric space.
topology
See topological space.
totally bounded
A metric space   is totally bounded if, for every  , there exist a finite cover of   by open balls of radius  . A metric space is compact if and only if it is complete and totally bounded.
totally disconnected
A space is totally disconnected if it has no connected subset with more than one point.
trivial topology
The trivial topology (or indiscrete topology) on a set   consists of precisely the empty set and the entire space  .
Tychonoff
A Tychonoff space (or completely regular Hausdorff space, completely   space,   space) is a completely regular   space. (A completely regular space is Hausdorff if and only if it is  , so the terminology is consistent.) Every Tychonoff space is regular Hausdorff.

U edit

ultra-connected
A space is ultra-connected if no two non-empty closed sets are disjoint. Every ultra-connected space is path-connected.
ultrametric
A metric is an ultrametric if it satisfies the following stronger version of the triangle inequality: for all   in  .
uniform isomorphism
If   and   are uniform spaces, a uniform isomorphism from   to   is a bijective function   such that   and   are uniformly continuous. The spaces are then said to be uniformly isomorphic and share the same uniform properties.
uniformisable
A space is uniformizable if it is homeomorphic to a uniform space.
uniform space
A uniform space is a set   equipped with a nonempty collection   of subsets of the Cartesian product   satisfying the following axioms:
  1. if   is in  , then   contains  .
  2. if   is in  , then   is also in  
  3. if   is in   and   is a subset of   which contains  , then   is in  
  4. if   and   are in  , then   is in  
  5. if   is in  , then there exists   in   such that, whenever   and   are in  , then   is in  .
The elements of   are called entourages, and   itself is called a uniform structure on  .
uniform structure
See uniform space.

W edit

weak topology
The weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the coarsest topology on the set which makes all the functions continuous.
weaker topology
See coarser topology. Beware, some authors, especially analysts, use the term stronger topology.
weakly countably compact
A space is weakly countably compact (or limit point compact) if every infinite subset has a limit point.
weakly hereditary
A property of spaces is said to be weakly hereditary if whenever a space has that property, then so does every closed subspace of it. For example, compactness and the Lindelöf property are both weakly hereditary properties, although neither is hereditary.
weight
The weight of a space   is the smallest cardinal number   such that   has a base of cardinal  . (Note that such a cardinal number exists, because the entire topology forms a base, and because the class of cardinal numbers is well-ordered.)
well-connected
See ultra-connected. (Some authors use this term strictly for ultra-connected compact spaces.)

Z edit

zero-dimensional
A space is zero-dimensional if it has a base of clopen sets.