Glossary of general topology

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no clear distinction between different areas of topology, this glossary focuses primarily on general topology and definitions that are fundamental to a broad range of areas. See the article on topological spaces for basic definitions and examples, and see the article on topology for a brief history and description of the subject area. See basic set theory and function for mathematical definitions concerning sets and functions.

The following articles may also be useful. These either contain specialised vocabulary within general topology or provide more detailed expositions of the definitions given below. The list of general topology topics will also be very helpful.

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

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A

Accessible. See T₁.

B

Baire space. A space is a Baire space if any intersection of countably many dense open sets is dense.

Base. A set of open sets is a base (or basis) for a topology if every open set in the topology is a union of sets in the base. The topology generated by a base is the smallest topology containing the base elements; this topology consists of all unions of elements of the base.

Basis. See Base.

Borel algebra. The Borel algebra on a space X is the smallest σ-algebra containing all the open sets.

Borel set. A Borel set is an element of a Borel algebra.

Boundary. The boundary 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.

C

Cauchy sequence. A sequence {x_i} in a metric space M with metric d is a Cauchy sequence (or Cauchy) if, for every positive real number r, there is an integer N such that for all integers m and n greater than N, the distance d(x_m, x_n) is less than r.

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

Closed set. A set is closed if its complement is a element 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 intersection of all closed sets which contain it. It is the smallest closed set containing the original set.

Closure operator. See Kuratowski closure axioms.

Coarser topology. If X is a set, and T₁ and T₂ are topologies on X, then T₁ is coarser (or smaller, weaker) than T₂ if T₁ is contained in T_{2_{. Beware, some authors, especially analysts, use the term stronger.}}

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.

Compact-open topology The compact-open topology on the set C(X, Y) of all continuous maps between two topological 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 Topologically complete.

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 p is a point not in C, then C and {p} are functionally separated.

Completely T₃. See Tychonoff.

Component. See connected component.

Connected. A space X 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 connected subspace. Each connected component is closed, and the set of connected components of a space forms 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 X is contractible if the identity map on X is homotopic to a constant map. Every contractible space is simply connected.

Countably compact. A space is countably compact if every countable open cover has a finite subcover.

Cover. A collection {U_i} of subsets of a space X is a cover (or covering) if their union is the whole space X.

Covering. See Cover.

D

Dense. A dense set is a set that meets every nonempty open set in the space. Equivalently, a set is dense if its closure is the whole space.

Discrete space. A space X is discrete if every subset of X is open. We say that X carries the discrete topology.

Discrete topology. See Discrete space.

E

Entourage. See Uniform space.

F

F_σ set. An F_σ set is a countable union of closed sets.

Finer topology. If X is a set, and T₁ and T₂ are topologies on X, then T₂ is finer (or stronger, larger) than T₁ if T₂ contains T_{1_{. Beware, some authors, especially analysts, use the term weaker.}}

First category. See Meagre.

First-countable. A space is first-countable if every point has a countable local base.

Functionally separated. Two sets A and B in a space X are functionally separated if there is a continuous function from X into the interval [0, 1] with the property that A is mapped to 0 and B is mapped to 1.

G

G_δ set. A G_δ set is a countable intersection of open sets.

H

Hausdorff. A space is Hausdorff (or T₂) if every two distinct points have disjoint neighbourhoods. Every Hausdorff space is T₁.

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. A homeomorphism from a space X to a space 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 speaking, this means that the space looks the same at every point. Every topological group is homogeneous.

Homotopic maps. Two continuous maps f, g : X -> Y are homotopic 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, the space X × [0, 1] is given the usual product topology. The function H is called a homotopy between f and g.

Homotopy. See Homotopic maps.

I

Identification space. See Quotient space.

Indiscrete space. See Trivial topology.

Indiscrete topology. See Trivial topology.

Interior. The interior of a set is the union of all open sets contained in it. It is the largest open set contained in the original set.

Isolated point. A point x is an isolated point if the singleton {x} is open.

K

Kolmogorov. See T₀.

Kuratowski closure axioms. The Kuratowski closure axioms is a set of axioms satisfied by the function which takes each subset of X to its closure:

Isotonicity: Every set is contained in its closure.
Idempotence: The closure of the closure of a set is equal to the closure of that set.
Preservation of binary unions: The closure of the union of two sets is the union of their closures.
Preservation of nullary unions: The closure of the empty set is empty.

If c is a function from the power set of X to itself, then 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 X by declaring the closed sets to be the fixed point of this operator, i.e. a set A is closed if and only if c(A) = A.

L

Larger topology. See Finer topology.

Limit point. A point x in 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.

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 topological 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 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 sets.

Locally finite. A collection of subsets of a space is locally finite if every point has a neighbourhood which meets only finitely many of the subsets.

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 sets. A locally path-connected space is connected if and only if it is path-connected.

M

Meagre. If X is a space and A is a subset of X, then A is meagre in X (or of first category in X) if it is the countable union of nowhere dense sets. If A is not meagre in X, A is sometimes said to be of second category in X.

Metric. See Metric space.

Metric space. A metric space is a set M equipped with a function d : M × M → R satisfying the following conditions 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. 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.

N

Neighbourhood/Neighborhood. A neighbourhood of a set S is a set containing an open set which in turn contains the set S. (Note that the neighbourhood itself need not be open.) A neighbourhood of a point p is a neighbourhood of the singleton set {p}.

Neighbourhood base. See Local base.

Neighbourhood basis. See Local base.

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 (x_α), where α is in 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. Every normal spaces admits a partition of unity.

Normal Hausdorff. A normal Hausdorff space (or T₄ space) is a normal T₁ space. (A normal space is Hausdorff if and only if it is T₁, so the terminology is consistent.) Normal Hausdorff spaces are Tychonoff.

Nowhere dense. A nowhere dense set is a set whose closure has empty interior.

O

Open cover. An open cover is a cover consisting of open sets.

Open set. A set is open if it 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.

P

Paracompact. A space is paracompact if every open cover has an open locally finite refinement. 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 topological space X is a continuous map from the closed unit interval [0, 1] into X.

Path-connected. A space X is path-connected if for every two points x, y in X, there is a path p from x to y, i.e., a continuous map p: [0,1] → X with p(0) = x and p(1) = y. Every path-connected spaces is connected.

Point. A point is an element of a topological space.

Polish. A space is called Polish if it is metrizable with a separable and complete metric.

Pre-compact. See Relatively compact.

Product topology. If {X_i} is a collection of spaces and X is the (set-theoretic) product of {X_i}, then the product topology on X is the coarsest topology for which all the projection maps are continuous.

Punctured neighbourhood/Punctured neighborhood. A punctured neighbourhood of a point p is a neighbourhood of p, minus {p}. For instance, the interval (−1, 1) = {x : −1 < x < 1} is a neighbourhood of 0 in the real line, so the set (−1, 0) ∪ (0, 1) = (−1, 1) − {0} is a punctured neighbourhood of 0.

Q

Quotient space. If X is a space, Y is a set, and f : X → Y is any surjective function, then the quotient space (or identification space) on Y induced by f is the finest topology for which f is continuous. 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.

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 p is a point not in C, then C and p 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.)

Relatively compact. A subset Y of a space X is relatively compact if the closure of Y in X is compact.

Residual. If X is a space and A is a subset of X, then A is residual in X if the complement of A is meagre in X.

S

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 separable, first-countable and Lindelöf.

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

Separated. Two sets A and B are separated if each is disjoint from the other's closure.

Sierpinski space. Let S = {0, 1}. Then T = {{}, {1}, {0, 1}} is a topology on S, and the resulting space is called the Sierpinski space. The Sierpinski space is the simplest example of a space that does not satisfy the T₁ axiom.

Simply connected. A space X is simply connected if it is path-connected and if every continuous map f: S¹ → X is homotopic to a constant map.

Smaller topology. See Coarser topology.

Stronger topology. See Finer topology. Beware, some authors, especially analysts, use the term weaker topology.

Subbase. A set of open sets is a subbase (or subbasis) for a topology if every open set in the topology is a union of finite intersections of sets in the subbase. The topology generated by a subbase is the coarsest topology containing the subbase elements; this topology consists of all finite intersections of unions of elements of the subbase.

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.

Subspace. If X is a space and A is a subset of X, then the subspace topology on A induced by X consists of all intersections of open sets in X with A.

Support. If X is a space and f : X → Y, where Y = the real numbers R or the complex numbers C, then the support of f is the closure of the set {x in X : f(x) ≠ 0}.

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 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.

T₃. See Regular Hausdorff.

T_3½. See Tychonoff.

T₄. See Normal Hausdorff.

T₅. See Completely normal Hausdorff.

Topological space. A topological space 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.

Topologically complete. A space is topologically complete if it is homeomorphic to a complete metric space.

Topology. See Topological space.

Topological vector space. A topological vector space is a real or complex vector space which is endowed with a Hausdorff topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous (where the product topologies are used and the base field K carries its standard topology).

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, T_3½ 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

Uniform space. A uniform space is a set U equipped with a nonempty collection Φ of subsets of the Cartesian product X × X satisfying the following axioms:

if U is in Φ, then U contains { (x, x) | x in X }.
if U is in Φ, then { (y, x) | (x, y) in U } is also in Φ
if U is in Φ and V is a subset of X × X which contains U, then V is in Φ
if U and V are in Φ, then U ∩ V is in Φ
if U is in Φ, then there exists V in Φ such that, whenever (x, y) and (y, z) are in V, then (x, z) is in U.

The elements of Φ are called entourages, and Φ itself is called a uniform structure on U.

Uniform structure. See Uniform space.

W

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 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.