Topology Glossary

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.

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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.

• 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.

• 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.

• 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}.

• 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.

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

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

• 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 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.

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Part 2 -- types of topological spaces

• 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 T₁ spaces are always Tychonoff.

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

• First-countable. A space is first-countable if at every point there is a countable set of open sets such that every neighbourhood of the point contains one of those open sets.

• Second-countable. A space is second-countable if it has countable base for its topology. Second-countable spaces are always separable.

• Compact. A space is compact if every open covering has a finite sub-covering. Compact Hausdorff spaces are always normal.

• Locally compact. A space is locally compact if every point has a compact neighborhood. Locally compact Hausdorff spaces are always Tychonoff.

• 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.

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

• 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.

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

• Paracompact. A space is paracompact if every open cover has an open locally finite refinement.

• Metrizable. A space is metrizable if it is homeomorphic to a metric space. Metrizable spaces are always Tychonoff, normal and paracompact.

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