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. For a list of terms specific to algebraic topology, see Glossary of algebraic topology.
All spaces in this glossary are assumed to be topological spaces unless stated otherwise.
A
Absolutely closed See H-closed
Accessible See .
Accumulation point See limit point.
Alexandroff compactification
Alexandrov topology The topology of a space X is an Alexandrov topology (or is finitely generated) if arbitrary intersections of open sets in X are open, or equivalently, if arbitrary unions of closed sets are closed, or, again equivalently, if the open sets are the upper sets of a poset.Vickers (1989) p.22
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.
α-closed, α-open A subset A of a topological space X is α-open if , and the complement of such a set is α-closed.
Approach space An approach space is a generalization of metric space based on point-to-set distances, instead of point-to-point.
B
Baire space This has two distinct common meanings: A space is a Baire space if the intersection of any countable collection of dense open sets is dense; see Baire space. 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 B 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.
β-open See Semi-preopen.
b-open, b-closed A subset of a topological space is b-open if . The complement of a b-open set is b-closed.
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 or .
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
Category of topological spaces The category Top has topological spaces as objects and continuous maps as morphisms.
Cauchy sequence A sequence {xn} in a metric space (M, d) is a Cauchy sequence if, for every positive real number r, there is an integer N such that for all integers m, n > N, we have d(xm, xn) < r.
Clopen set A set is clopen if it is both open and closed.
Closed ball If (M, d) is a metric space, a closed ball is a set of the form D(x; r) := {y in M : d(x, y) ≤ r}, where x is in M and r is a positive real number, the radius of the ball. A closed ball of radius r is a closed r-ball. Every closed ball is a closed set in the topology induced on M by d. Note that the closed ball D(x; r) might not be equal to the closure of the open ball 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 S is a point of closure of S.
Closure operator See Kuratowski closure axioms.
Coarser topology If X is a set, and if T1 and T2 are topologies on X, then T1 is coarser (or smaller, weaker) than T2 if T1 is contained in T2. Beware, some authors, especially analysts, use the term stronger.
Comeagre A subset A of a space X is comeagre (comeager) if its complement X\A is meagre. Also called residual.
Compact 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 C(X, Y) of all continuous maps between two spaces X and Y is defined as follows: given a compact subset K of X and an open subset U of Y, let V(K, U) denote the set of all maps f in C(X, Y) such that f(K) is contained in U. Then the collection of all such V(K, U) is a subbase for the compact-open topology.
Complete 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 T5 space) is a completely normal T1 space. (A completely normal space is Hausdorff if and only if it is T1, so the terminology is consistent.) Every completely normal Hausdorff space is normal Hausdorff.
Completely regular A space is completely regular if, whenever C is a closed set and x is a point not in C, then C and {x} are functionally separated.
Completely T3 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.
Continuum A space is called a continuum if it a compact, connected Hausdorff space.
Contractible A space X is contractible if the identity map on X is homotopic to a constant map. Every contractible space is simply connected.
Coproduct topology If {Xi} is a collection of spaces and X is the (set-theoretic) disjoint union of {Xi}, then the coproduct topology (or disjoint union topology, topological sum of the Xi) on X is the finest topology for which all the injection maps are continuous.
Core-compact space
Cosmic space A continuous image of some separable metric space.
Countable chain condition A space X satisfies the countable chain condition if every family of non-empty, pairswise disjoint open sets is countable.
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 X is countably locally finite (or σ-locally finite) if it is the union of a countable collection of locally finite collections of subsets of 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 X is a connected space with more than one point, then a point x of X is a cut point if the subspace X − {x} is disconnected.
D δ-cluster point, δ-closed, δ-open A point x of a topological space X is a δ-cluster point of a subset A if for every open neighborhood U of x in X. The subset A is δ-closed if it is equal to the set of its δ-cluster points, and δ-open if its complement is δ-closed.
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.
Dense-in-itself set A set is dense-in-itself if it has no isolated point.
Density the minimal cardinality of a dense subset of a topological space. A set of density ℵ0 is a separable space.Nagata (1985) p.104
Derived set If X is a space and S is a subset of X, the derived set of S in X is the set of limit points of S in X.
Developable space A topological space with a development.
Development A countable collection of open covers of a topological space, such that for any closed set C and any point p in its complement there exists a cover in the collection such that every neighbourhood of p in the cover is disjoint from C.
Diameter If (M, d) is a metric space and S is a subset of M, the diameter of S is the supremum of the distances d(x, y), where x and y range over S.
Discrete metric The discrete metric on a set X is the function d : X × X → R such that for all x, y in X, d(x, x) = 0 and d(x, y) = 1 if x ≠ y. The discrete metric induces the discrete topology on X.
Discrete space A space X is discrete if every subset of X is open. We say that X carries the discrete topology.Steen & Seebach (1978) p.41
Discrete topology See discrete space.
Disjoint union topology See Coproduct topology.
Dispersion point If X is a connected space with more than one point, then a point x of X is a dispersion point if the subspace X − {x} is hereditarily disconnected (its only connected components are the one-point sets).
Distance See metric space.
Dowker space
Dugundji extension theorem
Dunce hat (topology)
E
Entourage See Uniform space.
Exterior The exterior of a set is the interior of its complement.
Eilenberg–Wojdysławski theorem The Eilenberg–Wojdysławski theorem says every bounded metric space can be embedded into a Banach space as a closed subset of the convex hull of the image.
F
Fσ set An Fσ set is a countable union of closed sets.
Filter See also: Filters in topology. A filter on a space X is a nonempty family F of subsets of X such that the following conditions hold: The empty set is not in F. The intersection of any finite number of elements of F is again in F. If A is in F and if B contains A, then B is in F.
Final topology On a set X with respect to a family of functions into , is the finest topology on X which makes those functions continuous.
Fine topology (potential theory) On Euclidean space , the coarsest topology making all subharmonic functions (equivalently all superharmonic functions) continuous.
Finer topology If X is a set, and if T1 and T2 are topologies on X, then T2 is finer (or larger, stronger) than T1 if T2 contains T1. 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 T1.
Frontier See Boundary.
Full set A compact subset 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 A and B in a space X are functionally separated if there is a continuous map f: X → [0, 1] such that f(A) = 0 and f(B) = 1.
G
Gδ set A Gδ set or inner limiting set is a countable intersection of open sets.Steen & Seebach (1978) p.162
Gδ space A space in which every closed set is a Gδ set.
Generic point A generic point for a closed set is a point for which the closed set is the closure of the singleton set containing that point.Vickers (1989) p.65
H
Hausdorff A Hausdorff space (or T2 space) is one in which every two distinct points have disjoint neighbourhoods. Every Hausdorff space is T1. H-closed A space is H-closed, or Hausdorff closed or absolutely closed, if it is closed in every Hausdorff space containing it. Hemicompact A space is hemicompact, if there is a sequence of compact subsets so that every compact subset is contained in one of them.