Glossary of general 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. 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 <math>T_1</math>.

;Accumulation point: See limit point.

;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.

;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 <math>A \subseteq \operatorname{Int}_X \left( \operatorname{Cl}_X \left( \operatorname{Int}_X A \right) \right)</math>, 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 <math>\tau</math> if every open set in <math>\tau</math> is a union of sets in <math> B </math>. The topology <math>\tau</math> is the smallest topology on <math>X</math> containing <math>B</math> and is said to be generated by <math>B</math>.

;Basis: See Base.

;β-open: See Semi-preopen.

;b-open, b-closed: A subset <math>A</math> of a topological space <math>X</math> is b-open if <math>A \subseteq \operatorname{Int}_X \left( \operatorname{Cl}_X A \right) \cup \operatorname{Cl}_X \left( \operatorname{Int}_X A \right)</math>. The complement of a b-open set is b-closed.

;Borel algebra: The Borel algebra on a topological space <math> (X,\tau)</math> is the smallest <math>\sigma</math>-algebra containing all the open sets. It is obtained by taking intersection of all <math>\sigma</math>-algebras on <math> X </math> containing <math> \tau </math>.

;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 <math> A </math> is denoted by <math> \partial A</math> or <math>bd</math> <math>A</math>.

;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 <math>A \cap \operatorname{Int}_X\left( \operatorname{Cl}_X(U) \right) \neq \emptyset</math> 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.

;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.

;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.

;Fine topology (potential theory): On Euclidean space <math>\R^n</math>, 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.

;Gδ space: A space in which every closed set is a Gδ set.

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.

; Hereditarily P: A space is hereditarily P for some property P if every subspace is also P.

; 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 X and Y are spaces, a homeomorphism from X to Y is a bijective function f : X → Y such that f and f−1 are continuous. The spaces X and Y are then said to be homeomorphic. From the standpoint of topology, homeomorphic spaces are identical.

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

; Homotopic maps: Two continuous maps f, g : X  →  Y are homotopic (in Y) if there is a continuous map H : X × [0, 1]  →  Y such that H(x, 0) = f(x) and H(x, 1) = g(x) for all x in X. Here, X × [0, 1] is given the product topology. The function H is called a homotopy (in Y) between f and g.

; Homotopy: See Homotopic maps.

; Hyperconnected: A space is hyperconnected if no two non-empty open sets are disjoint

;Larger topology: See Finer topology.

;Limit point: A point x in a space X is a limit point of a subset S if every open set containing x also contains a point of S other than x itself. This is equivalent to requiring that every neighbourhood of x contains a point of S other than x itself.

;Limit point compact: See Weakly countably compact.

;Limit space

;Lindelöf: A space is Lindelöf if every open cover has a countable subcover.

;Local base: A set B of neighbourhoods of a point x of a space X is a local base (or local basis, neighbourhood base, neighbourhood basis) at x if every neighbourhood of x contains some member of B.

;Local basis: See Local base.

;Locally (P) space: There are two definitions for a space to be "locally (P)" where (P) is a topological or set-theoretic property: that each point has a neighbourhood with property (P), or that every point has a neighourbood base for which each member has property (P). The first definition is usually taken for locally compact, countably compact, metrizable, separable, countable; the second for locally connected.

;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 compact neighbourhood: the alternative definition that each point has a local base consisting of compact neighbourhoods is sometimes used: these are equivalent for Hausdorff spaces.

;Metacompact: A space is metacompact if every open cover has a point finite open refinement.

;Metric: See Metric space.

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

;Metric map: If X and Y are metric spaces with metrics dX and dY respectively, then a metric map is a function f from X to Y, such that for any points x and y in X, dY(f(x), f(y)) ≤ dX(x, y). A metric map is strictly metric if the above inequality is strict for all x and y in X.

;Metric space: A metric space (M, d) is a set M equipped with a function d : M × M → R satisfying the following axioms for all x, y, and z in M:

:# d(x, y) ≥ 0

:# d(x, x) = 0

:# if   d(x, y) = 0   then   x = y     (identity of indiscernibles)

:# d(x, y) = d(y, x)     (symmetry)

:# d(x, z) ≤ d(x, y) + d(y, z)     (triangle inequality)

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

;Moore space: A Moore space is a developable regular Hausdorff space.

N

; Nearly open: see preopen. See also: almost open map.

;Neighbourhood/Neighborhood: A neighbourhood of a point x is a set containing an open set which in turn contains the point x. More generally, a neighbourhood of a set S is a set containing an open set which in turn contains the set S. A neighbourhood of a point x is thus a neighbourhood of the singleton set {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 for a point x: A neighbourhood system at a point x in a space is the collection of all neighbourhoods of x.

;Net: A net in a space X is a map from a directed set A to X. A net from A to X is usually denoted (xα), where α is an index variable ranging over A. Every sequence is a net, taking 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. Paracompact Hausdorff spaces are normal.

;Partition of unity: A partition of unity of a space X is a set of continuous functions from X to [0, 1] 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 1.

;Path: A path in a space X is a continuous map f from the closed unit interval [0, 1] into X. The point f(0) is the initial point of f; the point f(1) is the terminal point of f.

;Path-connected: A space X is path-connected if, for every two points x, y in X, there is a path f from x to y, i.e., a path with initial point f(0) = x and terminal point f(1) = y. Every path-connected space is connected.

;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 completely metrizable, i.e. if it is homeomorphic to a separable and complete metric space.

;Polyadic: A space is polyadic if it is the continuous image of the power of a one-point compactification of a locally compact, non-compact Hausdorff space.

;Polytopological space: A polytopological space is a set <math>X</math> together with a family <math>\{\tau_i\}_{i\in I}</math> of topologies on <math>X</math> that is linearly ordered by the inclusion relation where <math>I</math> is an arbitrary index set.

;P-point: A point of a topological space is a P-point if its filter of neighbourhoods is closed under countable intersections.

;Pre-compact: See Relatively compact.

;: A subset A of a topological space X is preopen if <math>A \subseteq \operatorname{Int}_X \left( \operatorname{Cl}_X A \right)</math>.

;Prodiscrete topology: The prodiscrete topology on a product AG is the product topology when each factor A is given the discrete topology.

;Product topology: If <math>\left(X_i\right)</math> is a collection of spaces and X is the (set-theoretic) Cartesian product of <math>\left(X_i\right),</math> then the product topology on X is the coarsest topology for which all the projection maps are continuous.

;Proper function/mapping: A continuous function f from a space X to a space Y is proper if <math>f^{-1}(C)</math> is a compact set in X for any compact subspace C of Y.

;Proximity space: A proximity space (X, d) is a set X equipped with a binary relation d between subsets of X satisfying the following properties:

:For all subsets A, B and C of X,

:#A d B implies B d A

:#A d B implies A is non-empty

:#If A and B have non-empty intersection, then A d B

:#A d (B <math>\cup</math> C) if and only if (A d B or A d C)

:#If, for all subsets E of X, we have (A d E or B d E), then we must have A d (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 (M, d) is a set M equipped with a real-valued function <math>d : M \times M \to \R</math> 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 d is a pseudometric on M. Every metric is a pseudometric.

;Punctured neighbourhood/Punctured neighborhood: A punctured neighbourhood of a point x is a neighbourhood of x, minus {x}. For instance, the interval (−1, 1) = {y : −1 < y < 1} is a neighbourhood of x = 0 in the real line, so the set <math>(-1, 0) \cup (0, 1) = (-1, 1) - \{ 0 \}</math> is a punctured neighbourhood of 0.

Q

;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 X and Y are spaces, and if f is a surjection from X to Y, then f is a quotient map (or identification map) if, for every subset U of Y, U is open in Y if and only if f 1(U) is open in X. In other words, Y has the f-strong topology. Equivalently, <math>f</math> is a quotient map if and only if it is the transfinite composition of maps <math>X\rightarrow X/Z</math>, where <math>Z\subset X</math> is a subset. Note that this does not imply that f is an open function.

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

R

; Refinement: A cover K is a refinement of a cover L if every member of K is a subset of some member of L.

; Regular: A space is regular if, whenever C is a closed set and x is a point not in C, then C and x have disjoint neighbourhoods.

; Regular Hausdorff: A space is regular Hausdorff (or T3) if it is a regular T0 space. (A regular space is Hausdorff if and only if it is T0, so the terminology is consistent.)

; : A subset of a space X is regular open if it equals the interior of its closure; dually, a regular closed set is equal to the closure of its interior. An example of a non-regular open set is the set U = ∪ 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.

;Second category: See Meagre.

;Second-countable: A space is second-countable or perfectly separable if it has a countable base for its topology.

;Star: The star of a point in a given cover of a topological space is the union of all the sets in the cover that contain the point. See star refinement.

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

;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 the union of a finite intersection of sets in the subbase. If <math>\mathcal B</math> is any collection of subsets of a set X, the topology on X generated by <math>\mathcal B</math> is the smallest topology containing <math>\mathcal B;</math> this topology consists of the empty set, X and all unions of finite intersections of elements of <math>\mathcal B.</math> Thus <math>\mathcal B</math> is a subbase for the topology it generates.

;Subbasis: See Subbase.

;Subcover: A cover K is a subcover (or subcovering) of a cover L if every member of K is a member of L.

;Subcovering: See Subcover.

;Submaximal space: A topological space is said to be submaximal if every subset of it is locally closed, that is, every subset is the intersection of an open set and a closed set.

Here are some facts about submaximality as a property of topological spaces:

Every door space is submaximal.
Every submaximal space is weakly submaximal viz every finite set is locally closed.
Every submaximal space is irresolvable.

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

;Suslin line

T

;T0: A space is T0 (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.

;T1: A space is T1 (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 T0; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is T1 if all its singletons are closed. Every T1 space is T0.

;T2: See Hausdorff space.

;T3: See Regular Hausdorff.

;T3½: See Tychonoff space.

;T4: See Normal Hausdorff.

;T5: See Completely normal Hausdorff.

;Top: See Category of topological spaces.

;θ-cluster point, θ-closed, θ-open: A point x of a topological space X is a θ-cluster point of a subset A if <math>A \cap \operatorname{Cl}_X(U) \neq \emptyset</math> for every open neighborhood U of x in X. The subset A is θ-closed if it is equal to the set of its θ-cluster points, and θ-open if its complement is θ-closed.

;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 (X, T) is a set X equipped with a collection T of subsets of X satisfying the following axioms:

:# The empty set and X are in T.

:# The union of any collection of sets in T is also in T.

:# The intersection of any pair of sets in T is also in T.

:The collection T is a topology on X.

;Topological sum: See Coproduct topology.

;Topologically complete: Completely metrizable spaces (i. e. topological spaces homeomorphic to complete metric spaces) are often called topologically complete; sometimes the term is also used for Čech-complete spaces or completely uniformizable spaces.

;Topology: See Topological space.

;Totally bounded: A metric space M is totally bounded if, for every r > 0, there exist a finite cover of M by open balls of radius 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 X consists of precisely the empty set and the entire space X.

;Tychonoff: A Tychonoff space (or completely regular Hausdorff space, completely T3 space, T3.5 space) is a completely regular T0 space. (A completely regular space is Hausdorff if and only if it is T0, so the terminology is consistent.) Every Tychonoff space is regular Hausdorff.

U

;Ultra-connected: A space is ultra-connected if no two non-empty closed sets are disjoint.

References

Also available as Dover reprint.

External links

A glossary of definitions in topology