Appendix:Glossary of topology

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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.

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

Table of 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. — identity of indiscernibles
  2. — symmetry
  3. 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.