The topology of a space X is an
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, or, again equivalently, if the open sets are the
upper sets of a
poset.[1]
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.
α-closed, α-open
A subset A of a topological space X is α-open if , and the complement of such a set is α-closed.[2]
An
approach space is a generalization of metric space based on point-to-set distances, instead of point-to-point.
B
Baire space
This has two distinct common meanings:
A space is a Baire space if the intersection of any
countable collection of dense open sets is dense; see
Baire space.
Baire space is the set of all functions from the natural numbers to the natural numbers, with the topology of pointwise convergence; see
Baire space (set theory).
A collection B of open sets is a
base (or basis) for a topology if every open set in is a union of sets in . The topology is the smallest topology on containing and is said to be generated by .
The
Borel algebra on a topological space is the smallest
-algebra containing all the open sets. It is obtained by taking intersection of all -algebras on containing .
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. Boundary of a set is denoted by or .
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.
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
positivereal 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).
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.
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.
Comeagre
A subset A of a space X is comeagre (comeager) if its
complementX\A is
meagre. Also called residual.
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.
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.
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.
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.
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.
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.
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 (or σ-locally finite) if it is the union of a
countable collection of locally finite collections of subsets of X.
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
δ-cluster point, δ-closed, δ-open
A point x of a topological space X is a δ-cluster point of a subset A if for every open neighborhood U of x in X. The subset A is δ-closed if it is equal to the set of its δ-cluster points, and δ-open if its complement is δ-closed.[4]
A
countable collection of
open covers of a topological space, such that for any closed set C and any point p in its complement there exists a cover in the collection such that every neighbourhood of p in the cover is
disjoint from C.[6]
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.
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 hereditarily disconnected (its only connected components are the one-point sets).
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.
A property of spaces is said to be hereditary if whenever a space has that property, then so does every subspace of it.[12] For example, second-countability is a hereditary property.
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.
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.
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.
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.
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.
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.
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 ifc(A) = A.
Kolmogorov topology
TKol = {R, }∪{(a,∞): a is real number}; the pair (R,TKol) is named Kolmogorov Straight.
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.
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 (P) space
There are two definitions for a space to be "locally (P)" where (P) is a topological or set-theoretic property: that each point has a neighbourhood with property (P), or that every point has a neighourbood base for which each member has property (P). The first definition is usually taken for locally compact, countably compact, metrizable, separable, countable; the second for locally connected.[15]
A space is
locally compact if every point has a compact neighbourhood: the alternative definition that each point has a local base consisting of compact neighbourhoods is sometimes used: these are equivalent for Hausdorff spaces.[15] Every locally compact Hausdorff space is Tychonoff.
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, point finite.
Locally metrizable/Locally metrisable
A space is locally metrizable if every point has a metrizable neighbourhood.[15]
A space is
locally path-connected if every point has a local base consisting of path-connected neighbourhoods.[15] A locally path-connected space is connected
if and only if it is path-connected.
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 circleS1 into X.
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.[16]
If X and Y are metric spaces with metrics dX and dY respectively, then a
metric 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 metric map is
strictly metric if the above inequality is strict for all x and y in X.
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.
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.
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.)
A
net in a space X is a map from a
directed setA to X. A net from A to X is usually denoted (xα), 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.
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 space is Tychonoff.
An
open cover is a cover consisting of open sets.[6]
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
positivereal 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.
A function from one space to another is
open if the
image of every open set is open.
Open property
A property of points in a
topological space is said to be "open" if those points which possess it form an
open set. Such conditions often take a common form, and that form can be said to be an open condition; for example, in
metric spaces, one defines an open ball as above, and says that "strict inequality is an open condition".
A space is
paracompact if every open cover has a locally finite open refinement. Paracompact implies metacompact.[17] Paracompact Hausdorff spaces are normal.[18]
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.
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.[13]
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.[13]
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.[13] The set of path-connected components of a space X is denoted
π0(X).
A collection B of nonempty open sets is a π-base for a topology τ if every nonempty open set in τ includes a set from B.[19]
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".
If is a collection of spaces and X is the (set-theoretic)
Cartesian product of 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 is a compact set in X for any compact subspace C of Y.
A pseudometric space (M, d) is a set M equipped with a
real-valued function satisfying all the conditions of a metric space, except possibly the identity of indiscernibles. That is, points in a pseudometric space may be "infinitely close" without being identical. 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 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.
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 Yif and only iff-1(U) is open in X. In other words, Y has the f-strong topology. Equivalently, is a quotient map if and only if it is the transfinite composition of maps , where is a subset. Note that this does not imply that f is an open function.
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.
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.)
A subset of a space X is regular open if it equals the interior of its closure; dually, a regular closed set is equal to the closure of its interior.[21] An example of a non-regular open set is the set U = (0,1) ∪ (1,2) in R with its normal topology, since 1 is in the interior of the closure of U, but not in U. The regular open subsets of a space form a
complete Boolean algebra.[21]
A space is
second-countable or perfectly separable if it has a
countable base for its topology.[8] Every second-countable space is first-countable, separable, and Lindelöf.
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.)
Semi-open
A subset A of a topological space X is called semi-open if .[23]
Semi-preopen
A subset A of a topological space X is called semi-preopen if [2]
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.
A collection of open sets is a
subbase (or subbasis) for a topology if every non-empty proper open set in the topology is the union of a
finite intersection of sets in the subbase. If is any collection of subsets of a set X, the topology on X generated by is the smallest topology containing this topology consists of the empty set, X and all unions of finite intersections of elements of Thus is a subbase for the topology it generates.
A
topological space is said to be submaximal if every subset of it is locally closed, that is, every subset is the intersection of an
open set and a
closed set.
Here are some facts about submaximality as a property of topological spaces:
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.
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.
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.
A point x of a topological space X is a θ-cluster point of a subset A if for every open neighborhood U of x in X. The subset A is θ-closed if it is equal to the set of its θ-cluster points, and θ-open if its complement is θ-closed.[23]
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.
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.
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.[13] Every ultra-connected space is path-connected.
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)).
If X and Y are
uniform spaces, a uniform isomorphism from X to Y is a bijective function f : X → Y such that f and f−1 are
uniformly continuous. The spaces are then said to be uniformly isomorphic and share the same
uniform properties.
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 U ∩ V 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 X. The uniform structure induces a topology on X where the basic neighborhoods of x are sets of the form {y : (x,y)∈U} for U∈Φ.
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.
Weight
The
weight of a spaceX is the smallest
cardinal number κ such that X has a base of cardinal κ. (Note that such a cardinal number exists, because the entire topology forms a base, and because the class of cardinal numbers is
well-ordered.)
Well-connected
See Ultra-connected. (Some authors use this term strictly for ultra-connected compact spaces.)
^
abGabbay, Dov M.; Kanamori, Akihiro; Woods, John Hayden, eds. (2012). Sets and Extensions in the Twentieth Century. Elsevier. p. 290.
ISBN978-0444516213.
The topology of a space X is an
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, or, again equivalently, if the open sets are the
upper sets of a
poset.[1]
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.
α-closed, α-open
A subset A of a topological space X is α-open if , and the complement of such a set is α-closed.[2]
An
approach space is a generalization of metric space based on point-to-set distances, instead of point-to-point.
B
Baire space
This has two distinct common meanings:
A space is a Baire space if the intersection of any
countable collection of dense open sets is dense; see
Baire space.
Baire space is the set of all functions from the natural numbers to the natural numbers, with the topology of pointwise convergence; see
Baire space (set theory).
A collection B of open sets is a
base (or basis) for a topology if every open set in is a union of sets in . The topology is the smallest topology on containing and is said to be generated by .
The
Borel algebra on a topological space is the smallest
-algebra containing all the open sets. It is obtained by taking intersection of all -algebras on containing .
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. Boundary of a set is denoted by or .
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.
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
positivereal 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).
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.
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.
Comeagre
A subset A of a space X is comeagre (comeager) if its
complementX\A is
meagre. Also called residual.
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.
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.
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.
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.
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.
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.
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 (or σ-locally finite) if it is the union of a
countable collection of locally finite collections of subsets of X.
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
δ-cluster point, δ-closed, δ-open
A point x of a topological space X is a δ-cluster point of a subset A if for every open neighborhood U of x in X. The subset A is δ-closed if it is equal to the set of its δ-cluster points, and δ-open if its complement is δ-closed.[4]
A
countable collection of
open covers of a topological space, such that for any closed set C and any point p in its complement there exists a cover in the collection such that every neighbourhood of p in the cover is
disjoint from C.[6]
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.
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 hereditarily disconnected (its only connected components are the one-point sets).
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.
A property of spaces is said to be hereditary if whenever a space has that property, then so does every subspace of it.[12] For example, second-countability is a hereditary property.
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.
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.
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.
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.
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.
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.
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 ifc(A) = A.
Kolmogorov topology
TKol = {R, }∪{(a,∞): a is real number}; the pair (R,TKol) is named Kolmogorov Straight.
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.
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 (P) space
There are two definitions for a space to be "locally (P)" where (P) is a topological or set-theoretic property: that each point has a neighbourhood with property (P), or that every point has a neighourbood base for which each member has property (P). The first definition is usually taken for locally compact, countably compact, metrizable, separable, countable; the second for locally connected.[15]
A space is
locally compact if every point has a compact neighbourhood: the alternative definition that each point has a local base consisting of compact neighbourhoods is sometimes used: these are equivalent for Hausdorff spaces.[15] Every locally compact Hausdorff space is Tychonoff.
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, point finite.
Locally metrizable/Locally metrisable
A space is locally metrizable if every point has a metrizable neighbourhood.[15]
A space is
locally path-connected if every point has a local base consisting of path-connected neighbourhoods.[15] A locally path-connected space is connected
if and only if it is path-connected.
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 circleS1 into X.
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.[16]
If X and Y are metric spaces with metrics dX and dY respectively, then a
metric 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 metric map is
strictly metric if the above inequality is strict for all x and y in X.
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.
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.
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.)
A
net in a space X is a map from a
directed setA to X. A net from A to X is usually denoted (xα), 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.
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 space is Tychonoff.
An
open cover is a cover consisting of open sets.[6]
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
positivereal 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.
A function from one space to another is
open if the
image of every open set is open.
Open property
A property of points in a
topological space is said to be "open" if those points which possess it form an
open set. Such conditions often take a common form, and that form can be said to be an open condition; for example, in
metric spaces, one defines an open ball as above, and says that "strict inequality is an open condition".
A space is
paracompact if every open cover has a locally finite open refinement. Paracompact implies metacompact.[17] Paracompact Hausdorff spaces are normal.[18]
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.
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.[13]
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.[13]
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.[13] The set of path-connected components of a space X is denoted
π0(X).
A collection B of nonempty open sets is a π-base for a topology τ if every nonempty open set in τ includes a set from B.[19]
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".
If is a collection of spaces and X is the (set-theoretic)
Cartesian product of 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 is a compact set in X for any compact subspace C of Y.
A pseudometric space (M, d) is a set M equipped with a
real-valued function satisfying all the conditions of a metric space, except possibly the identity of indiscernibles. That is, points in a pseudometric space may be "infinitely close" without being identical. 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 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.
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 Yif and only iff-1(U) is open in X. In other words, Y has the f-strong topology. Equivalently, is a quotient map if and only if it is the transfinite composition of maps , where is a subset. Note that this does not imply that f is an open function.
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.
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.)
A subset of a space X is regular open if it equals the interior of its closure; dually, a regular closed set is equal to the closure of its interior.[21] An example of a non-regular open set is the set U = (0,1) ∪ (1,2) in R with its normal topology, since 1 is in the interior of the closure of U, but not in U. The regular open subsets of a space form a
complete Boolean algebra.[21]
A space is
second-countable or perfectly separable if it has a
countable base for its topology.[8] Every second-countable space is first-countable, separable, and Lindelöf.
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.)
Semi-open
A subset A of a topological space X is called semi-open if .[23]
Semi-preopen
A subset A of a topological space X is called semi-preopen if [2]
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.
A collection of open sets is a
subbase (or subbasis) for a topology if every non-empty proper open set in the topology is the union of a
finite intersection of sets in the subbase. If is any collection of subsets of a set X, the topology on X generated by is the smallest topology containing this topology consists of the empty set, X and all unions of finite intersections of elements of Thus is a subbase for the topology it generates.
A
topological space is said to be submaximal if every subset of it is locally closed, that is, every subset is the intersection of an
open set and a
closed set.
Here are some facts about submaximality as a property of topological spaces:
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.
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.
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.
A point x of a topological space X is a θ-cluster point of a subset A if for every open neighborhood U of x in X. The subset A is θ-closed if it is equal to the set of its θ-cluster points, and θ-open if its complement is θ-closed.[23]
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.
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.
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.[13] Every ultra-connected space is path-connected.
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)).
If X and Y are
uniform spaces, a uniform isomorphism from X to Y is a bijective function f : X → Y such that f and f−1 are
uniformly continuous. The spaces are then said to be uniformly isomorphic and share the same
uniform properties.
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 U ∩ V 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 X. The uniform structure induces a topology on X where the basic neighborhoods of x are sets of the form {y : (x,y)∈U} for U∈Φ.
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.
Weight
The
weight of a spaceX is the smallest
cardinal number κ such that X has a base of cardinal κ. (Note that such a cardinal number exists, because the entire topology forms a base, and because the class of cardinal numbers is
well-ordered.)
Well-connected
See Ultra-connected. (Some authors use this term strictly for ultra-connected compact spaces.)
^
abGabbay, Dov M.; Kanamori, Akihiro; Woods, John Hayden, eds. (2012). Sets and Extensions in the Twentieth Century. Elsevier. p. 290.
ISBN978-0444516213.