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# Topology glossary

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.

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.

• Accessible. See T1.

• 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.

• Cauchy sequence. A sequence {xi} in a metric space M with metric d is called a Cauchy sequence (or Cauchy for short) 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(xm, xn) is less than r.

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

• 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 intersection of all closed sets which contain it. It is the smallest closed set containing the original set.

• 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.

• 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.) Completely normal Hausdorff spaces are always 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. The connected components of a space form 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. Contractible spaces are always simply connected.

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

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

• Covering. See Cover.

• 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.

• Entourage. See Uniform space.

• Fσ set. An Fσ set is a countable union of closed sets.

• 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.

• Gδ set. A Gδ set is a countable intersection of open sets.

• Hausdorff. A space is Hausdorff (or T2) if every two distinct points have disjoint neighbourhoods. Hausdorff spaces are always 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 map fX → 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. All topological groups are homogeneous.

• Homotopic maps. Two continuous maps f, g : X -> Y are homotopic if there is a continuous map HX× [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.

• 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.

• Kolmogorov. See T0.

• Kuratowski closure axioms. The Kuratowski closure axioms are a set of axioms satisied by the closure operator:
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.

• 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.

• Lindel�f. A space is Lindel�f if every open cover has a countable subcover. See open cover.

• 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 neighbourhoodss. Locally compact Hausdorff spaces are always 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.

• 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 dM × 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 called a metric on M.

• Metrizable/Metrisable. 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.

• 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 (xα), 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. Normal spaces admit partitions 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 always Tychonoff.

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

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

• 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.

• 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-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.

• Point. This term is often used to refer to elements of the topological space.

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

• Product topology. If {Xi} is a collection of spaces and X is the (set-theoretic) product of {Xi}, then the product topology on X is the weakest 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.

• Quotient space. If X and Y are spaces and fX → Y is any function, then the quotient space on Y induced by f is the weakest 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.

• 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.

• Second category. See Meagre.

• 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.

• 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 = is a topology on S, and the resulting space is called 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 every continuous map f: S1 → X is homotopic to a constant map.

• 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 smallest 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.

• 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. T1 spaces are always T0.

• T2. See Hausdorff.

• T3. See Regular Hausdorff.

• T. 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 conditions:

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 called a topology on X.

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

• Topology. See Topological space.

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

• Trivial topology. The trivial 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 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.) Tychonoff spaces are always regular Hausdorff.

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

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 UV 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.

• Weak topology. The weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the weakest topology on the set which makes all the functions continuous.

• 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.