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Topology glossary

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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.

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 Naïve set theory, Axiomatic set theory, and Function for 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 and List of examples in general topology will also be very helpful.

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

A

• Accessible. See T1.
• Alexandrov topology. A space X has the 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.
• 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.

B

• Base. A collection B of open sets is a base (or basis) for a topology T if every open set in T is a union of sets in B. The topology T is the smallest topology on X containing B and is said to be generated by B.
• 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.
• 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

• 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 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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 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

• 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.
• 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.
• Diameter. If (M, d) is a metric space and S is a subset of M, the diameter of S is the supremum of the distances d(x, y), where x and y range over S.
• Discrete metric. The discrete metric on a set X is the function d : X × X  →  R such that for all x, y in X, d(x, x) = 0 and d(x, y) = 1 if xy. The discrete metric induces the discrete topology on X.
• Discrete space. A space X is discrete if every subset of X is open. We say that X carries the discrete topology.
• 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 totally disconnected.

E

• Exterior. The exterior of a set is the interior of its complement.

F

• Filter. A filter on a space X is a nonempty family F of subsets of X such that the following conditions hold:
1. The empty set is not in F.
2. The intersection of any finite number of elements of F is again in F.
3. If A is in F and if B contains A, then B is in F.
• 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.
• Fréchet. See T1.
• 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.

H

• 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.
• Hyper-connected. A space is hyper-connected if no two non-empty open sets are disjoint. Every hyper-connected space is connected.

I

• Identification map. See Quotient map.
• Interior. The interior of a set is the largest open set contained in the original set. It is equal to the union of all open sets contained in it. An element of the interior of a set S is an interior point of S.
• Isolated point. A point x is an isolated point if the singleton {x} is open. More generally, if S is a subset of a space X, and if x is a point of S, then x is an isolated point of S if {x} is open in the subspace topology on S.
• Isometric isomorphism. If M1 and M2 are metric spaces, an isometric isomorphism from M1 to M2 is a bijective isometry f : M1  →  M2. The metric spaces are then said to be isometrically isomorphic. From the standpoint of metric space theory, isometrically isomorphic spaces are identical.
• Isometry. If (M1, d1) and (M2, d2) are metric spaces, an isometry from M1 to M2 is a function f : M1  →  M2 such that d2(f(x), f(y)) = d1(x, y) for all x, y in M1. Every isometry is injective, although not every isometry is surjective.

K

1. Isotonicity : Every set is contained in its closure.
2. Idempotence: The closure of the closure of a set is equal to the closure of that set.
3. Preservation of binary unions: The closure of the union of two sets is the union of their closures.
4. 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 points of this operator, i.e. a set A is closed if and only if c(A) = A.

L

• 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.
• 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 closed subset. A subset of a topological space that is the intersection of an open and a closed subset.
• 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 finite. A collection of subsets of a space is locally finite if every point has a neighbourhood which has nonempty intersection with only finitely many of the subsets. See also countably locally finite.
• Locally metrizable/Locally metrisable. A space is locally metrizable if every point has a metrizable neighbourhood.
• Locally simply connected. A space is locally simply connected if every point has a local base consisting of simply connected neighbourhoods.
• Loop. If x is a point in a space X, a loop at x in X (or a loop in X with basepoint x) is a path f in X, such that f(0) = f(1) = x. Equivalently, a loop in X is a continuous map from the unit circle S1 into X.

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 of second category in X.
• Metric invariant. A metric invariant is a property which is preserved under isometric isomorphism.
• 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:
1. d(x, y) ≥ 0
2. d(x, x) = 0
3. if   d(x, y) = 0   then   x = y     (identity of indiscernibles)
4. d(x, y) = d(y, x)     (symmetry)
5. 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.

N

• 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.)
• Normal. A space is normal if any two disjoint closed sets have disjoint neighbourhoods. Every normal space admits a partition of unity.

O

• Open ball. If (M, d) is a metric space, an open ball is a set of the form B(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. An open ball of radius r is an open r-ball. Every open ball is an open set in the topology on M induced by d.

P

• Paracompact. A space is paracompact if every open cover has a locally finite open 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 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.
• Path-connected component. A path-connected component of a space is a maximal nonempty path-connected subspace. The set of path-connected components of a space is a partition of that space, which is finer than the partition into connected components. The set of path-connected components of a space X is denoted π0(X).
• 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".
• Polish. A space is Polish if it is separable and topologically complete, i.e. if it is homeomorphic to a separable and complete metric space.
• 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 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 f−1(C) is a compact set in X for any compact subspace C of Y.
• Proximity space A proximity space (Xδ) is a set X equipped with a relation δ between subsets of X satisfying the following properties:
For all subsets A, B and C of X,
1. A δ B implies B δ A
2. A δ B implies A is non-empty
3. If A and B have non-empty intersection, then A δ B
4. A δ (B ∪ C) iff (A δ B or A δ C)
5. If, for all subsets E of X, we have (A δ E or B δ E), then we must have A δ (XB)
• 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 function d : M × M → R satisfying all the conditions of a metric space, except possibly the identity of indiscernibles. 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 (−1, 0) ∪ (0, 1) = (−1, 1) − {0} 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.
• 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 open. An open subset U of a space X is regular open if it equals the interior of its closure. The regular open subsets of a space form a complete Boolean algebra.
• 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.
• Semilocally simply connected. A space X is semilocally simply connected if, for every point x in X, there is a neighbourhood U of x such that every loop at x in U is homotopic in X to the constant loop x. Every simply connected space and every locally simply connected space is semilocally simply connected. (Compare with locally simply connected; here, the homotopy is allowed to live in X, whereas in the definition of locally simply connected, the homotopy must live in U.)
• Sequentially compact. A space is sequentially compact if every sequence has a convergent subsequence. Every sequentially compact space is countably compact, and every first-countable, countably compact space is sequentially compact.
• Short map. If X and Y are metric spaces with metrics dX and dY respectively, then a short 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 short map is strictly short if the above inequality is strict for all x and y in X.
• 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 open set in the topology is a union of finite intersections of sets in the subbase. If B is any collection of subsets of a set X, the topology on X generated by B is the smallest topology containing B; this topology consists of all unions of finite intersections of elements of B.
• 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 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.

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.
• 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.
1. The empty set and X are in T.
2. The union of any collection of sets in T is also in T.
3. 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.
• 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.
• 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. Every ultra-connected space is path-connected.
• Ultrametric. A metric is an ultrametric if it satisfies the following stronger version of the triangle inequality: for all x, y, z in M, d(x, z) ≤ max(d(x, y), d(y, z)).
• Uniformizable/Uniformisable. A space is uniformizable if it is homeomorphic to a uniform space.
1. if U is in Φ, then U contains { (x, x) | x in X }.
2. if U is in Φ, then { (y, x) | (x, y) in U } is also in Φ
3. if U is in Φ and V is a subset of X × X which contains U, then V is in Φ
4. if U and V are in Φ, then UV is in Φ
5. 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.

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.
• Weakly countably compact. A space is weakly countably compact (or limit point compact) if every infinite subset has a limit point.
• 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.
• Well-connected. See Ultra-connected. (Some authors use this term strictly for ultra-connected compact spaces.)

Z

• Zero-dimensional. A space is zero-dimensional if it has a base of clopen sets.

Last updated: 05-15-2005 09:29:55
03-10-2013 05:06:04