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:

Glossary of topology

Wikipedia

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

accessible

See ${\displaystyle T_{1}}$.

accumulation point

See limit point.

Alexandrov topology

A space ${\displaystyle X}$ has the Alexandrov topology (or is finitely generated) if arbitrary intersections of open sets in ${\displaystyle X}$ 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.

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 ${\displaystyle B}$ of open sets is a base (or basis) for a topology ${\displaystyle \tau }$ if every open set in ${\displaystyle \tau }$ is a union of sets in ${\displaystyle B}$. The topology ${\displaystyle \tau }$ is the smallest topology on ${\displaystyle X}$ containing ${\displaystyle B}$ and is said to be generated by${\displaystyle B}$.

basis

See base.

Borel algebra

The Borel algebra on a topological space ${\displaystyle (X,\tau )}$ is the smallest ${\displaystyle \sigma }$-algebra containing all the open sets. It is obtained by taking intersection of all ${\displaystyle \sigma }$-algebras on ${\displaystyle X}$ containing ${\displaystyle \tau }$.

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 ${\displaystyle A}$ is denoted by ${\displaystyle \partial A}$

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.

category of topological spaces

The category Top has topological spaces as objects and continuous maps as morphisms.

Cauchy sequence

A sequence ${\displaystyle x_{n}}$ in a metric space ${\displaystyle (M,d)}$ is a Cauchy sequence if, for every positive real number ${\displaystyle r}$, there is an integer ${\displaystyle N}$ such that for all integers ${\displaystyle m,n>N}$, we have ${\displaystyle d(x_{m},x_{n})<r}$.

clopen set

A set is clopen if it is both open and closed.

closed ball

If ${\displaystyle (M,d)}$ is a metric space, a closed ball is a set of the form ${\displaystyle D(x;r):=\{y{\text{ in }}M:d(x,y)\leq r\}}$, where ${\displaystyle x}$ is in ${\displaystyle M}$ and ${\displaystyle r}$ is a positive real number, the radius of the ball. A closed ball of radius ${\displaystyle r}$ is a closed ${\displaystyle r}$-ball. Every closed ball is a closed set in the topology induced on ${\displaystyle M}$ by ${\displaystyle d}$. Note that the closed ball ${\displaystyle D(x;r)}$ might not be equal to the closure of the open ball ${\displaystyle B(x;r)}$.

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 ${\displaystyle S}$ is a point of closure of ${\displaystyle S}$.

closure operator

See Kuratowski closure axioms.

coarser topology

If ${\displaystyle X}$ is a set, and if ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$ are topologies on ${\displaystyle X}$, then ${\displaystyle T_{1}}$ is coarser (or smaller, weaker) than ${\displaystyle T_{2}}$ if ${\displaystyle T_{1}}$ is contained in ${\displaystyle T_{2}}$. Beware, some authors, especially analysts, use the term stronger.

comeagre

A subset ${\displaystyle A}$ of a space ${\displaystyle X}$ is comeagre (comeager) if its complement ${\displaystyle X\backslash A}$ 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 ${\displaystyle C(X,Y)}$ of all continuous maps between two spaces ${\displaystyle X}$ and ${\displaystyle Y}$ is defined as follows: given a compact subset ${\displaystyle K}$ of ${\displaystyle X}$ and an open subset ${\displaystyle U}$ of ${\displaystyle Y}$, let ${\displaystyle V(K,U)}$ denote the set of all maps ${\displaystyle f}$ in ${\displaystyle C(X,Y)}$ such that ${\displaystyle f(K)}$ is contained in ${\displaystyle (U)}$. Then the collection of all such ${\displaystyle V(K,U)}$ is a subbase for the compact-open topology.

complete, complete space

A metric space is complete if every Cauchy sequence converges.

completely metrizable/-isable

See complete space.

completely normal

A space is completely normal if any two separated sets have disjoint neighborhoods.

completely normal Hausdorff

A completely normal Hausdorff space (or ${\displaystyle T_{5}}$ space) is a completely normal ${\displaystyle T_{1}}$ space. (A completely normal space is Hausdorff if and only if it is ${\displaystyle T_{1}}$, so the terminology is consistent.) Every completely normal Hausdorff space is normal Hausdorff.

completely regular

A space is completely regular if, whenever ${\displaystyle C}$ is a closed set and ${\displaystyle x}$ is a point not in ${\displaystyle C}$, then ${\displaystyle C}$ and ${\displaystyle \{x\}}$ are functionally separated.

completely ${\displaystyle T_{3}}$

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 ${\displaystyle X}$ is contractible if the identity map on ${\displaystyle X}$ is homotopic to a constant map. Every contractible space is simply connected.

coproduct topology

If ${\displaystyle \{X_{i}\}}$ is a collection of spaces and ${\displaystyle X}$ is the (set-theoretic) disjoint union of ${\displaystyle \{X_{i}\}}$, then the coproduct topology (or disjoint union topology, topological sum of the ${\displaystyle \{X_{i}\}}$ on ${\displaystyle X}$ 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 ${\displaystyle X}$ is countably locally finite (or ${\displaystyle \sigma }$-locally finite) if it is the union of a countable collection of locally finite collections of subsets of ${\displaystyle X}$.

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 ${\displaystyle X}$ is a connected space with more than one point, then a point ${\displaystyle x}$ of ${\displaystyle X}$ is a cut point if the subspace ${\displaystyle X-\{x\}}$ is disconnected.

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 ${\displaystyle X}$ is a space and ${\displaystyle S}$ is a subset of ${\displaystyle X}$, the derived set of ${\displaystyle S}$ in ${\displaystyle X}$ is the set of limit points of ${\displaystyle S}$ in ${\displaystyle X}$.

diameter

If ${\displaystyle (M,d)}$ is a metric space and ${\displaystyle S}$ is a subset of ${\displaystyle M}$, the diameter of ${\displaystyle S}$ is the supremum of the distances ${\displaystyle d(x,y)}$, where ${\displaystyle x}$ and ${\displaystyle y}$ range over ${\displaystyle S}$.

discrete metric

The discrete metric on a set ${\displaystyle X}$ is the function ${\displaystyle d:X\times X\rightarrow }$ ; R such that for all ${\displaystyle x}$, ${\displaystyle y}$ in ${\displaystyle X}$, ${\displaystyle d(x,x)=0}$ and ${\displaystyle d(x,y)=1}$ if ${\displaystyle x\neq y}$. The discrete metric induces the discrete topology on X.

discrete space

A space ${\displaystyle X}$' is discrete if every subset of ${\displaystyle X}$ is open. We say that ${\displaystyle X}$ carries the discrete topology.

discrete topology

See discrete space.

disjoint union topology

See coproduct topology.

dispersion point

If ${\displaystyle X}$ is a connected space with more than one point, then a point ${\displaystyle x}$ of ${\displaystyle X}$ is a dispersion point if the subspace ${\displaystyle X-\{x\}}$ is hereditarily disconnected (its only connected components are the one-point sets).

distance

See metric space.

dunce hat

entourage

See uniform space.

exterior

The exterior of a set is the interior of its complement.

${\displaystyle F_{\sigma }}$ set

An ${\displaystyle F_{\sigma }}$ set is a countable union of closed sets.

filter

A filter on a space ${\displaystyle X}$ is a nonempty family ${\displaystyle F}$ of subsets of ${\displaystyle X}$ such that the following conditions hold:

1. The empty set is not in ${\displaystyle F}$.
2. The intersection of any finite number of elements of ${\displaystyle F}$ is again in ${\displaystyle F}$.
3. If ${\displaystyle A}$ is in ${\displaystyle F}$ and if ${\displaystyle B}$ contains ${\displaystyle A}$, then ${\displaystyle B}$ is in ${\displaystyle F}$.

finer topology

If ${\displaystyle X}$ is a set, and if ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$ are topologies on ${\displaystyle X}$, then ${\displaystyle T_{2}}$ is finer (or larger, stronger) than ${\displaystyle T_{1}}$ if ${\displaystyle T_{2}}$ contains ${\displaystyle T_{1}}$. 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 ${\displaystyle T_{1}}$.

frontier

See boundary.

full set

A compact subset ${\displaystyle K}$ 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 ${\displaystyle A}$ and ${\displaystyle B}$ in a space ${\displaystyle X}$ are functionally separated if there is a continuous map ${\displaystyle f:X\rightarrow }$ such that ${\displaystyle f(A)=0}$ and ${\displaystyle f(B)=1}$.

${\displaystyle G_{\delta }}$ set

A ${\displaystyle G_{\delta }}$ set is a countable intersection of open sets.

Hausdorff

A Hausdorff space (or ${\displaystyle T_{2}}$ space) is one in which every two distinct points have disjoint neighbourhoods. Every Hausdorff space is ${\displaystyle T_{1}}$.

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 ${\displaystyle X}$ and ${\displaystyle Y}$ are spaces, a homeomorphism from ${\displaystyle X}$ to ${\displaystyle Y}$ is a bijective function ${\displaystyle f:X\rightarrow Y}$ such that ${\displaystyle f}$ and ${\displaystyle f^{-1}}$ are continuous. The spaces ${\displaystyle X}$ and ${\displaystyle Y}$ are then said to be homeomorphic. From the standpoint of topology, homeomorphic spaces are identical.

homogeneous

A space ${\displaystyle X}$ is homogeneous if, for every ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle X}$, there is a homeomorphism ${\displaystyle f:X\rightarrow }$; ${\displaystyle X}$ such that ${\displaystyle f(x)=y}$. Intuitively, the space looks the same at every point. Every topological group is homogeneous.

homotopic, homotopic maps

Two continuous maps ${\displaystyle f,g:X\rightarrow Y}$ are homotopic (in ${\displaystyle Y}$) if there is a continuous map ${\displaystyle H:X\times \rightarrow Y\rightarrow Y}$ such that ${\displaystyle H(x,0)=f(x)}$ and ${\displaystyle H(x,1)=g(x)}$ for all ${\displaystyle x}$ in ${\displaystyle X}$. Here, ${\displaystyle X\times }$ is given the product topology. The function ${\displaystyle H}$ is called a homotopy (in ${\displaystyle Y}$) between ${\displaystyle f}$ and ${\displaystyle g}$.

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.

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 ${\displaystyle S}$ is an interior point of ${\displaystyle S}$.

interior point

See interior.

isolated point

A point ${\displaystyle x}$ is an isolated point if the singleton ${\displaystyle \{x\}}$ is open. More generally, if ${\displaystyle S}$ is a subset of a space ${\displaystyle X}$, and if ${\displaystyle x}$ is a point of ${\displaystyle S}$, then ${\displaystyle x}$ is an isolated point of ${\displaystyle S}$ if ${\displaystyle \{x\}}$ is open in the subspace topology on ${\displaystyle S}$.

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.

Kolmogorov axiom

See ${\displaystyle T_{0}}$.

Kuratowski closure axioms

The Kuratowski closure axioms is a set of axioms satisfied by the function which takes each subset of ${\displaystyle X}$ 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 ${\displaystyle c}$ is a function from the power set of ${\displaystyle X}$ to itself, then ${\displaystyle c}$ is a closure operator if it satisfies the Kuratowski closure axioms. The Kuratowski closure axioms can then be used to define a topology on ${\displaystyle X}$ by declaring the closed sets to be the fixed points of this operator, i.e. a set ${\displaystyle A}$ is closed if and only if ${\displaystyle c(A)=A}$.

larger topology

See finer topology.

limit point

A point ${\displaystyle x}$ in a space ${\displaystyle X}$ is a limit point of a subset ${\displaystyle S}$ if every open set containing ${\displaystyle x}$ also contains a point of ${\displaystyle S}$ other than ${\displaystyle x}$ itself. This is equivalent to requiring that every neighbourhood of ${\displaystyle x}$ contains a point of ${\displaystyle S}$ other than ${\displaystyle x}$ 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 ${\displaystyle B}$ of neighbourhoods of a point ${\displaystyle x}$ of a space ${\displaystyle X}$ is a local base (or local basis, neighbourhood base, neighbourhood basis) at ${\displaystyle x}$ if every neighbourhood of ${\displaystyle x}$ contains some member of ${\displaystyle B}$.

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 neighborhood which has nonempty intersection with only finitely many of the subsets. See also countably locally finite, point finite.

locally metrizable/-isable

A space is locally metrizable if every point has a metrizable neighborhood.

locally path-connected

A space is locally path-connected if every point has a local base consisting of path-connected neighborhoods. 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 neighborhoods.

loop

If ${\displaystyle x}$ is a point in a space ${\displaystyle X}$, a loop at ${\displaystyle x}$ in ${\displaystyle X}$ (or a loop in ${\displaystyle X}$, with basepoint ${\displaystyle x}$) is a path ${\displaystyle f}$ in ${\displaystyle X}$, such that ${\displaystyle f(0)=f(1)=x}$. Equivalently, a loop in ${\displaystyle X}$ is a continuous map from the unit circle ${\displaystyle S^{1}}$, into ${\displaystyle X}$.

meagre

If ${\displaystyle X}$ is a space and ${\displaystyle A}$ is a subset of ${\displaystyle X}$, then ${\displaystyle A}$ is meagre in ${\displaystyle X}$ (or of first category in ${\displaystyle X}$) if it is the countable union of nowhere dense sets. If ${\displaystyle A}$ is not meagre in ${\displaystyle X}$, ${\displaystyle A}$ is of second category in ${\displaystyle X}$.

metric

See metric space.

metric invariant

A metric invariant is a property which is preserved under isometric isomorphism.

metric map

If ${\displaystyle X}$ and ${\displaystyle Y}$ are metric spaces with metrics ${\displaystyle d_{X}}$ and ${\displaystyle d_{Y}}$ respectively, then a metric map is a function ${\displaystyle f}$ from ${\displaystyle X}$ to ${\displaystyle Y}$, such that for any points ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle X}$, ${\displaystyle d_{Y}(f(x),f(y))\leq d_{X}(x,y)}$. A metric map is strictly metric if the above inequality is strict for all ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle X}$.

metric space

A metric space ${\displaystyle (M,d)}$ is a set ${\displaystyle M}$ equipped with a function ${\displaystyle d:M\times M\rightarrow }$ ${\displaystyle \mathbb {R} }$ satisfying the following axioms for all ${\displaystyle x}$, ${\displaystyle y}$ and ${\displaystyle z}$ in ${\displaystyle M}$:

1. ${\displaystyle d(x,y)\geq 0}$
2. ${\displaystyle d(x,x)=0}$
3. ${\displaystyle {\text{if }}d(x,y)=0{\text{ then }}x=y}$ — identity of indiscernibles
4. ${\displaystyle d(x,y)=d(y,x)}$ — symmetry
5. ${\displaystyle d(x,z)\leq d(x,y)+d(y,z)}$ — triangle inequality

The function ${\displaystyle d}$ is a metric on ${\displaystyle M}$, and ${\displaystyle d(x,y)}$ is the distance between ${\displaystyle x}$ and ${\displaystyle y}$. The collection of all open balls of M is a base for a topology on ${\displaystyle M}$; this is the topology on ${\displaystyle M}$ induced by ${\displaystyle d}$. 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 ${\displaystyle X}$ has a largest proper open subset; this subset is called a monolith.

neighbourhood/neighborhood

A neighbourhood of a point ${\displaystyle x}$ is a set containing an open set which in turn contains the point ${\displaystyle x}$. More generally, a neighbourhood of a set ${\displaystyle S}$ is a set containing an open set which in turn contains the set ${\displaystyle S}$. A neighbourhood of a point ${\displaystyle x}$ is thus a neighbourhood of the singleton set ${\displaystyle \{x\}}$. (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 ${\displaystyle x}$ in a space is the collection of all neighbourhoods of ${\displaystyle x}$.

net

A net in a space ${\displaystyle X}$ is a map from a directed set ${\displaystyle A}$ to ${\displaystyle X}$. A net from ${\displaystyle A}$ to ${\displaystyle X}$ is usually denoted (${\displaystyle x_{\alpha }}$), where ${\displaystyle \alpha }$ is an index variable ranging over ${\displaystyle A}$. Every sequence is a net, taking ${\displaystyle A}$ 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 ${\displaystyle T_{4}}$ space) is a normal ${\displaystyle T_{1}}$ space. (A normal space is Hausdorff if and only if it is ${\displaystyle T_{1}}$, 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.

open cover

An open cover is a cover consisting of open sets.

open ball

If ${\displaystyle (M,d)}$ is a metric space, an open ball is a set of the form ${\displaystyle B(x;r):=\{y{\text{ in }}M:d(x,y)<r\}}$, where ${\displaystyle x}$ is in ${\displaystyle M}$ and ${\displaystyle r}$ is a positive real number, the radius of the ball. An open ball of radius ${\displaystyle r}$ is an open ${\displaystyle r}$-ball. Every open ball is an open set in the topology on ${\displaystyle M}$ induced by ${\displaystyle d}$.

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".

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 ${\displaystyle X}$ is a set of continuous functions from ${\displaystyle X}$ to ${\displaystyle }$ 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 ${\displaystyle 1}$.

path

A path in a space ${\displaystyle X}$ is a continuous map ${\displaystyle f}$ from the closed unit interval into ${\displaystyle X}$. The point ${\displaystyle f(0)}$ is the initial point of ${\displaystyle f}$; the point ${\displaystyle f(1)}$ is the terminal point of ${\displaystyle f}$.

path-connected

A space ${\displaystyle X}$ is path-connected if, for every two points ${\displaystyle x}$, ${\displaystyle y}$ in ${\displaystyle X}$, there is a path ${\displaystyle f}$ from ${\displaystyle x}$ to ${\displaystyle y}$, i.e., a path with initial point ${\displaystyle f(0)=x}$ and terminal point ${\displaystyle f(1)=y}$. 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 ${\displaystyle \pi _{0}(X)}$.

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 ${\displaystyle \{X_{i}\}}$ is a collection of spaces and ${\displaystyle X}$ is the (set-theoretic) product of ${\displaystyle \{X_{i}\}}$, then the product topology on ${\displaystyle X}$ is the coarsest topology for which all the projection maps are continuous.

proper function/mapping

A continuous function f from a space ${\displaystyle X}$ to a space ${\displaystyle Y}$ is proper if ${\displaystyle f^{-1}(C)}$ is a compact set in ${\displaystyle X}$ for any compact subspace ${\displaystyle C}$ of ${\displaystyle Y}$.

proximity space

A proximity space ${\displaystyle (X,\mathbf {\delta } )}$ is a set ${\displaystyle X}$ equipped with a binary relation ${\displaystyle \mathbf {\delta } }$ between subsets of ${\displaystyle X}$ satisfying the following properties:

For all subsets ${\displaystyle A}$, ${\displaystyle B}$ and ${\displaystyle C}$ of ${\displaystyle X}$,

1. ${\displaystyle A\mathbf {\delta } B}$ implies ${\displaystyle B\mathbf {\delta } A}$
2. ${\displaystyle A\mathbf {\delta } B}$ implies ${\displaystyle A}$ is non-empty
3. If ${\displaystyle A}$ and ${\displaystyle B}$ have non-empty intersection, then ${\displaystyle (A\mathbf {\delta } B)}$
4. ${\displaystyle A\mathbf {\delta } (B\cup C)}$ iff (${\displaystyle A\mathbf {\delta } B}$ or ${\displaystyle A\mathbf {\delta } C}$)
5. If, for all subsets E of X, we have (${\displaystyle A\mathbf {\delta } E}$ or ${\displaystyle B\mathbf {\delta } E}$), then we must have ${\displaystyle A\mathbf {\delta } (X-B)}$

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 ${\displaystyle (M,d)}$ is a set ${\displaystyle M}$ equipped with a function ${\displaystyle d:M\times M\rightarrow }$ ${\displaystyle \mathbb {R} }$ 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 ${\displaystyle d}$ is a pseudometric on ${\displaystyle M}$. Every metric is a pseudometric.

punctured neighbourhood/punctured neighborhood

A punctured neighbourhood of a point ${\displaystyle x}$ is a neighbourhood of ${\displaystyle x}$, minus ${\displaystyle \{x\}}$. For instance, the interval ${\displaystyle (-1,1)=\{y:-1<y<1\}}$ is a neighbourhood of ${\displaystyle x=0}$ in the real line, so the set ${\displaystyle (-1,0)\cup (0,1)=(-1,1)-\{0\}}$ is a punctured neighbourhood of ${\displaystyle 0}$.

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 ${\displaystyle X}$ and ${\displaystyle Y}$ are spaces, and if ${\displaystyle f}$ is a surjection from ${\displaystyle X}$ to ${\displaystyle Y}$, then ${\displaystyle f}$ is a quotient map (or identification map) if, for every subset ${\displaystyle U}$ of ${\displaystyle Y}$, ${\displaystyle U}$ is open in ${\displaystyle Y}$ if and only if ${\displaystyle f^{-1}(U)}$ is open in ${\displaystyle X}$. In other words, ${\displaystyle Y}$ has the ${\displaystyle f}$-strong topology. Equivalently, ${\displaystyle f}$ is a quotient map if and only if it is the transfinite composition of maps ${\displaystyle X\rightarrow X/Z}$, where ${\displaystyle Z\subset X}$ is a subset. Note that this doesn't imply that f is an open function.

quotient space

If ${\displaystyle X}$ is a space, ${\displaystyle Y}$ is a set, and ${\displaystyle f:X\rightarrow Y}$ is any surjective function, then the quotient topology on ${\displaystyle Y}$ induced by ${\displaystyle f}$ is the finest topology for which ${\displaystyle f}$ is continuous. The space ${\displaystyle X}$ is a quotient space or identification space. By definition, ${\displaystyle f}$ is a quotient map. The most common example of this is to consider an equivalence relation on ${\displaystyle X}$, with ${\displaystyle Y}$ the set of equivalence classes and ${\displaystyle f}$ the natural projection map. This construction is dual to the construction of the subspace topology.

refinement

A cover ${\displaystyle K}$ is a refinement of a cover ${\displaystyle L}$ if every member of ${\displaystyle K}$ is a subset of some member of ${\displaystyle L}$.

regular

A space is regular if, whenever ${\displaystyle C}$ is a closed set and ${\displaystyle x}$ is a point not in ${\displaystyle C}$, then ${\displaystyle C}$ and ${\displaystyle x}$ have disjoint neighbourhoods.

regular Hausdorff

A space is regular Hausdorff (or ${\displaystyle T_{3}}$) if it is a regular ${\displaystyle T_{0}}$ space. (A regular space is Hausdorff if and only if it is ${\displaystyle T_{0}}$, so the terminology is consistent.)

regular open

An open subset ${\displaystyle U}$ of a space ${\displaystyle X}$ is regular open if it equals the interior of its closure. An example of a non-regular open set is the set ${\displaystyle U=(0,1)\cup (1,2){\text{ in }}R}$ 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 ${\displaystyle Y}$ of a space ${\displaystyle X}$ is relatively compact in ${\displaystyle X}$ if the closure of ${\displaystyle Y}$ in ${\displaystyle X}$ is compact.

residual

If ${\displaystyle X}$ is a space and ${\displaystyle A}$ is a subset of ${\displaystyle X}$, then ${\displaystyle A}$ is residual in ${\displaystyle X}$ if the complement of ${\displaystyle A}$ is meagre in ${\displaystyle X}$. Also called comeagre or comeager.

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 ${\displaystyle X}$ is semilocally simply connected if, for every point ${\displaystyle x}$ in ${\displaystyle X}$, there is a neighbourhood ${\displaystyle U}$ of ${\displaystyle x}$ such that every loop at ${\displaystyle x}$ in ${\displaystyle U}$ is homotopic in ${\displaystyle X}$ to the constant loop ${\displaystyle X}$. 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 ${\displaystyle X}$, whereas in the definition of locally simply connected, the homotopy must live in ${\displaystyle U}$.)

separable

A space is separable if it has a countable dense subset.

separated

Two sets ${\displaystyle A}$ and ${\displaystyle B}$ 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.

${\displaystyle f}$-Strong topology

Let ${\displaystyle f\colon X\rightarrow Y}$ be a map of topological spaces. We say that ${\displaystyle Y}$ has the ${\displaystyle f}$-strong topology if, for every subset ${\displaystyle U\subset Y}$, one has that ${\displaystyle U}$ is open in ${\displaystyle Y}$ if and only if ${\displaystyle f^{-1}(U)}$ is open in ${\displaystyle X}$

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 ${\displaystyle B}$ is any collection of subsets of a set ${\displaystyle X}$, the topology on ${\displaystyle X}$ generated by ${\displaystyle B}$ is the smallest topology containing ${\displaystyle B}$; this topology consists of the empty set, ${\displaystyle X}$ and all unions of finite intersections of elements of ${\displaystyle B}$.

subbasis

See subbase.

subcover

A cover ${\displaystyle K}$ is a subcover (or subcovering) of a cover ${\displaystyle L}$ if every member of ${\displaystyle K}$ is a member of ${\displaystyle L}$.

subcovering

See subcover.

subspace

If T is a topology on a space ${\displaystyle X}$, and if ${\displaystyle A}$ is a subset of ${\displaystyle X}$, then the subspace topology on ${\displaystyle A}$ induced by ${\displaystyle T}$ consists of all intersections of open sets in ${\displaystyle T}$ with ${\displaystyle A}$. This construction is dual to the construction of the quotient topology.

${\displaystyle T_{0}}$

A space is ${\displaystyle T_{0}}$ (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.

${\displaystyle T_{1}}$

A space is ${\displaystyle T_{1}}$ (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 ${\displaystyle T_{0}}$; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is ${\displaystyle T_{1}}$ if all its singletons are closed. Every ${\displaystyle T_{1}}$ space is ${\displaystyle T_{0}}$.

${\displaystyle T_{2}}$

See Hausdorff space.

${\displaystyle T_{3}}$

See regular Hausdorff.

${\displaystyle T_{3.5}}$

See Tychonoff space.

${\displaystyle T_{4}}$

See normal Hausdorff.

${\displaystyle T_{5}}$

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 ${\displaystyle (X,T)}$ is a set ${\displaystyle X}$ equipped with a collection ${\displaystyle T}$ of subsets of ${\displaystyle X}$ satisfying the following axioms:

1. The empty set and ${\displaystyle X}$ are in ${\displaystyle T}$.
2. The union of any collection of sets in ${\displaystyle T}$ is also in ${\displaystyle T}$.
3. The intersection of any pair of sets in ${\displaystyle T}$ is also in ${\displaystyle T}$.

The collection ${\displaystyle T}$ is a topology on ${\displaystyle X}$.

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 ${\displaystyle M}$ is totally bounded if, for every ${\displaystyle r>0}$, there exist a finite cover of ${\displaystyle M}$ by open balls of radius ${\displaystyle r}$. 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 ${\displaystyle X}$ consists of precisely the empty set and the entire space ${\displaystyle X}$.

Tychonoff

A Tychonoff space (or completely regular Hausdorff space, completely ${\displaystyle T_{3}}$ space, ${\displaystyle T_{3.5}}$ space) is a completely regular ${\displaystyle T_{0}}$ space. (A completely regular space is Hausdorff if and only if it is ${\displaystyle T_{0}}$, so the terminology is consistent.) Every Tychonoff space is regular Hausdorff.

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 ${\displaystyle x,y,z}$ in ${\displaystyle M,d(x,z)\leq max(d(x,y),d(y,z))}$.

uniform isomorphism

If ${\displaystyle X}$ and ${\displaystyle Y}$ are uniform spaces, a uniform isomorphism from ${\displaystyle X}$ to ${\displaystyle Y}$ is a bijective function ${\displaystyle f:X\rightarrow Y}$ such that ${\displaystyle f}$ and ${\displaystyle f^{-1}}$ 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 ${\displaystyle U}$ equipped with a nonempty collection ${\displaystyle \Phi }$ of subsets of the Cartesian product ${\displaystyle X\times X}$ satisfying the following axioms:

1. if ${\displaystyle U}$ is in ${\displaystyle \Phi }$, then ${\displaystyle U}$ contains ${\displaystyle \{(x,x)|x{\text{ in}}X\}}$.
2. if ${\displaystyle U}$ is in ${\displaystyle \Phi }$, then ${\displaystyle \{(y,x)|(x,y){\text{ in}}U\}}$ is also in ${\displaystyle \Phi }$
3. if ${\displaystyle U}$ is in ${\displaystyle \Phi }$ and ${\displaystyle V}$ is a subset of ${\displaystyle X\times X}$ which contains ${\displaystyle U}$, then ${\displaystyle V}$ is in ${\displaystyle \Phi }$
4. if ${\displaystyle U}$ and ${\displaystyle V}$ are in ${\displaystyle \Phi }$, then ${\displaystyle U\cap V}$ is in ${\displaystyle \Phi }$
5. if ${\displaystyle U}$ is in ${\displaystyle \Phi }$, then there exists ${\displaystyle V}$ in ${\displaystyle \Phi }$ such that, whenever ${\displaystyle (x,y)}$ and ${\displaystyle (y,z)}$ are in ${\displaystyle V}$, then ${\displaystyle (x,z)}$ is in ${\displaystyle (U)}$.

The elements of ${\displaystyle \Phi }$ are called entourages, and ${\displaystyle \Phi }$ itself is called a uniform structure on ${\displaystyle U}$.

uniform structure

See uniform space.

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 ${\displaystyle X}$ is the smallest cardinal number ${\displaystyle \kappa }$ such that ${\displaystyle X}$ has a base of cardinal ${\displaystyle \kappa }$. (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.)

zero-dimensional

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