2 B

• Baire space. A space is a Baire space if the intersection of any countable collection of dense open sets is dense.

• 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.

• Basis. See Base.

• Borel algebra. The Borel algebra on a space 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 (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.

3 C

• Category of topological spaces. The category Top has topological spaces as objects and continuous maps as morphisms.

• Cauchy sequence. A sequence {x_n} in a metric space (M, d) is a Cauchy sequence if, for every positive real number r, there is an integer N such that for all integers m, n > N, we have d(x_m, x_n) < r.

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

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

• Closure operator. See Kuratowski closure axioms.

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

• Completely T₃. See Tychonoff.

• Component. See Connected component/Path-connected component.

• 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.

• Connected component. A connected component of a space is a maximal nonempty connected subspace. Each connected component is closed, and the set of connected components of a space is 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.

• Coproduct topology. If {X_i} is a collection of spaces and X is the (set-theoretic) disjoint union of {X_i}, then the coproduct topology (or disjoint union topology, topological sum of the X_i) 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.

• 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.

4 D

• Dense. 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 x ≠ y. 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.

• Discrete topology. See Discrete space.

• Disjoint union topology. See Coproduct topology.

• Distance. See Metric space.

5 E

• Entourage. See Uniform space.

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

6 F

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

• 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 T_{1. Beware, some authors, especially analysts, use the term weaker.}

• Finitely generated. See Alexandrov topology.

• First category. See Meagre.

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

• Fréchet. See T₁.

• Frontier. See Boundary.

• 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.

7 G

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

8 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. 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.

• Homotopy. See Homotopic maps.

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

9 I

• Identification map. See Quotient map.

• Identification space. See Quotient space.

• Indiscrete space. See Trivial topology.

• Indiscrete topology. See Trivial topology.

• 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.

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

• 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.

10 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:

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.

11 L

• Larger topology. See Finer topology.

• 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.

• 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 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 connected. A space is locally connected if every point has a local base consisting of connected neighbourhoods.

• Locally simply connected. A space is locally simply connected if every point has a local base consisting of simply connected neighbourhoods.

• 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.

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

• 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.

12 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. See Metric space.

• 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. The largest open proper subset of a non-empty ultra-connected compact space.

13 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 under this definition, the neighbourhood itself need not be open. Many authors require that neighbourhoods be open; be careful to note conventions.) A neighbourhood of a point x is a neighbourhood of the singleton set {x}.

• Neighbourhood base/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 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 space 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.) Every normal Hausdorff spaces is Tychonoff.

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

14 O

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

• 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.

• Open set. An open set 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.

15 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.

• 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.

• 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.

• 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.

• 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 indiscernables. 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.

16 Q

• 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.

17 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 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 in X 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.

18 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 first-countable, separable, and Lindelöf.

• 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.)

• 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.

• 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.

• Simply connected. A space is simply connected if it is path-connected and every loop 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 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.

• 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 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.

19 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.

• T₂. See Hausdorff.

• T₃. See Regular Hausdorff.

• T_3½. See Tychonoff.

• T₄. See Normal Hausdorff.

• T₅. See Completely normal Hausdorff.

• Top. See Category of topological spaces.

• 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.

• Topological space. A topological space (X, T) is a set X equipped with a collection T of subsets of X satisfying the following axioms:

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.

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

• Topology. See Topological space.

• 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.

• 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.

20 U

• Ultra-connected. A space is ultra-connected if no two non-empty closed sets are disjoint. Every ultra-connected space is path-connected.

• Ultra-connected compact. In an ultra-connected compact space X, every open cover must contain X. Every non-empty ultra-connected compact space has a largest proper open subset called a monolith.

• 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)).

• 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:

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 U ∩ V 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.

• Uniform structure. See Uniform space.

21 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 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 and Ultra-connected compact.

22 Z

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

Topology General topology Topology

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