Glossary of general topology

This is a glossary of some terms used in the branch of mathematics known as topology. See the article on topology for basic definitions.

This glossary is divided into two parts. The first part deals with general concepts, and the second part lists types of topological spaces defined in terms of these concepts. All spaces in this glossary are assumed to be topological spaces.

Part 1 -- topological concepts

Continuous. A function from one space to another is continuous if the preimage of every open set is open.

Homeomorphic. Two spaces X and Y are homeomorphic if there is a bijective map f : X -> Y such that f and f^-1 are continuous. From the standpoint of topology, X and Y are the same. The function f is called a homeomorphism.

Closure. The closure of a set is the intersection of all closed sets which contain it. It is the smallest closed set containing the original set.

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.

Boundary. The boundary of a set is the set's closure minus its interior.

Dense. A dense set is a set whose closure is the whole space.

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

Neighbourhood. 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 p is a neighbourhood of the 1-point set {p}.

Punctured neighbourhood. A punctured neighbourhood of a point p is a neighbourhood of p, minus p. For instance, {x: 0<|x|<1} is a punctured neighbourhood of 0 in the real line, because {x : 0 ≤ |x| < 1} is a neighbourhood of 0.

Sub-base. A set of open sets is a sub-base for a topology if every open set is a union of finite intersections of sets in the sub-base.

Base, or Basis. A set of open sets is a base for a topology if every open set is a union of sets in the base.

Local base. A set B of neighbourhoods of a point x of a topological space X is a local base at x if every neighbourhood of x contains some member of B.

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.

Cover. A collection {U_i} of sets is a cover (or covering), if their union is the whole space. An open cover is a cover {U_i} in which each U_i is an open set.

Subcover. A cover K is a subcover of a cover L if every member of K is a member of L.

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

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

Partition of unity. A partition of unity is a set of continuous functions from a space to [0,1] such that any point has a neighbourhood where all but a finite number are identically zero, and the sum of all them at every point is 1.

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. The function h is called a homotopy between f and g.

Part 2 -- types of topological spaces

Topological spaces can be classified regarding the degree to which their points are separated, regarding their compactness, their overall size and their connectedness.

Separation axioms

For a detailed treatment, see Separation axiom. Some of these terms are defined differently in older mathematical literature; see History of the separation axioms.

T₀. A space is T₀ if for every pair of distinct points in the space, there is an open set containing one but not the other.

T₁. A space is T₁ if all its singletons are closed. T₁ spaces are always T₀.

Hausdorff. A space is Hausdorff if every two distinct points have disjoint neighborhoods. Hausdorff spaces are always T₁.

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 T₀ spaces are always Hausdorff.

Tychonoff. A Hausdorff space is Tychonoff if whenever C is a closed set and p is a point not in C, then C and p are functionally separated. Tychonoff spaces are always regular.

Normal. A space is normal if any two disjoint closed sets have disjoint neighbourhoods. Normal spaces admit partitions of unity. Normal T₁ spaces are always Tychonoff.

Compactness

Paracompact. A space is paracompact if every open cover has an open locally finite refinement. Paracompact Hausdorff spaces are normal.

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

Compact. A space is compact if every open cover has a finite subcover. Compact spaces are always Lindelöf and paracompact. Compact Hausdorff spaces are therefore normal.

Locally compact. A space is locally compact if every point has a local base consisting of compact neighborhoods. Locally compact Hausdorff spaces are always Tychonoff.

Size

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

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

Second-countable. A space is second-countable if it has a countable base for its topology. Second-countable spaces are always separable, first-countable and Lindelöf.

Connectedness

Connected. A space X is connected if it is not the union of a pair of disjoint nonempty open sets.

Totally disconnected. A space is totally disconnected if it has no connected subset with more than one point.

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. Path-connected spaces are always connected.

Locally path-connected. A space is locally path-connected if every point has a local base consisting of open path-connected sets. A Locally path-connected space is connected if and only if it is path-connected.

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

Contractible. A space X is contractible if the identity map on X is homotopic to a constant map. Contractible spaces are always simply connected.

Miscellaneous

Metrizable. A space is metrizable if it is homeomorphic to a metric space. Metrizable spaces are always Hausdorff and paracompact (and hence normal and Tychonoff), and first-countable.

Locally metrizable. A space is locally metrizable if every point has a metrizable neighbourhood.

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. All topological groups are homogeneous.