# Topological space

A totality of two elements: A set $X$, consisting of elements of an arbitrary nature, called points of the given space, and a topological structure, or topology, on this set $X$ (cf. Topological structure (topology)); it is immaterial whether this is an open or closed topology (one transfers into the other by replacing the sets constituting the given topology by their complements). Unless otherwise stated, a topology $\mathfrak{G}$ will be assumed to be open. Logically the simplest method of specifying a topology on a given set $X$ consists in directly indicating those subsets of $X$ that make up this topology. But it is often simpler to specify not all the sets making up the given topology but only some of them (i.e. to specify a base of the given topology), in terms of which all remaining elements of the topology can be obtained as unions (in the case of an open topology) or intersections (in the closed case) of sets belonging to the base. So, for example, the usual topology of the real line is obtained by taking as a base of its open topology the set of all open intervals (it is sufficient to take only open intervals with rational end points). The remaining open sets are unions of such intervals.

When regarding a base of an open, or closed, topology, it is common to refer to it as an open or closed base of the given topological space. Open bases are more often considered than closed ones, hence if one speaks simply of a base of a topological space, an open base is meant. The smallest (in non-trivial cases, infinite) cardinal number that is the cardinality of a base of a given topological space is called its weight (cf. Weight of a topological space). After the cardinality of the set of all its points, the weight is the most important so-called cardinal invariant of the space (see Cardinal characteristic). Of special importance are spaces having a countable base, for example the real line. One obtains a countable base of an $n$-dimensional Euclidean space by taking the so-called rational (open) balls, that is, balls for which the radius and the coordinates of the centre are rational numbers. A topology is often defined by some (natural) standard procedure on a set equipped with some other structure. Thus one speaks of the natural topology of a metric space or of the natural (interval) topology of a totally ordered set. The former has as base the set of all open balls of the given metric space, the latter has as base the open intervals of the given totally ordered set.

A topological space is said to be metrizable (see Metrizable space) if there is a metric on its underlying set which induces the given topology. The metrizable spaces form one of the most important classes of topological spaces, and for several decades some of the central problems in general topology were the general and special problems of metrization, i.e. problems on finding necessary and sufficient conditions for a topological space, or for a topological space of some particular type, to be metrizable. These conditions form the content of general or special metrization theorems.

Every subset $X_0$ of the set $X$ of all points of a given topological space $(X,\mathfrak{G})$ can naturally be made into a topological space (a subspace of $(X,\mathfrak{G})$), by taking the topology consisting of all possible sets of the form $X_0\cap \Gamma$, where $\Gamma$ is an arbitrary element of $\mathfrak{G}$. Since a topology is a set of subsets of $X$, there is a natural (partial) order relation (by inclusion) between different topologies on a given set $X$, i.e. a topology $\mathfrak{G}_2$ is greater than (or equal to) a topology $\mathfrak{G}_1$ if $\mathfrak{G}_1$ is a subset of the set $\mathfrak{G}_2$, i.e. if each set that is open in the topology $\mathfrak{G}_1$ is open in the topology $\mathfrak{G}_2$. If it is clear from the context which topology is being used on a given set $X$, one simply denotes the topological space $(X,\mathfrak{G})$ by $X$. From the concept of a topology all the remaining fundamental topological concepts may be derived. First of all, closed sets are defined as complements of open sets. Moreover, any open set containing a point $x$ of a given space $X$ is said to be a neighbourhood of $x$. The concept of a neighbourhood enables one to define a closure point or proximate point of a set $M\subset X$ as a point every neighbourhood of which has non-empty intersection with $M$. This definition implies that every point of the set $M$ itself is a proximate point of $M$. The set of all proximate points of $M$ is called the closure of $M$ and is denoted by $[M]$ (cf. Closure of a set). The transition from an arbitrary set $M$ to its closure is called the closure operation in the given topological space. Properties of this operation are: 1) $M\subset [M]$, and $M= [M]$ if and only if $M$ is closed, i.e. if its complement is open; 2) $[M_1\cup M_2]=[M_1]\cup [M_2]$; and 3) $[\,[M]\,]=[M]$. The closure of an arbitrary set $M$ is the intersection of all closed sets containing $M$; alternatively, $[M]$ is the smallest closed set containing $M$.

The closure operation and its basic properties 1), 2) and 3) have been derived from the fundamental concept of a topology on a given set $X$. Alternatively, it is possible to consider as fundamental topological concept that of closure. I.e., to assume that in a given abstract set $X$ for each subset $M$ there is specified a subset $[M]$, called the closure of $M$, such that the properties 1), 2) and 3) are satisfied (these are then called axioms of closure or Kuratowski axioms), together with 4) $\emptyset=[\,\emptyset\,]$. In this approach closed sets are defined as sets coinciding with their closures, and open sets are defined as complements of closed ones. Thus one obtains precisely a topology in the initial sense; moreover, the closure operation to which it leads coincides with the one given a priori. This approach was chosen by K. Kuratowski (1922) in order to construct the concept of a topological space. In 1925 open topological structures were introduced by P.S. Aleksandrov. Both approaches lead to the same class of topological space that is currently the most generally accepted one.

Closely associated with the concept of a topological space is that of a continuous mapping from one space into another. A mapping $f:X\to Y$ from a topological space $X$ into a topological space $Y$ is continuous at a point $x\in X$ if for an arbitrary neighbourhood $O_y$ of the point $y=f(x)\in Y$ there exists a neighbourhood $O_x$ of $x$ in $X$ such that $f(O_x)\subset O_y$ (the Cauchy condition). Without altering the content of the definition, it is possible to take the neighbourhoods $O_y$ and $O_x$ to belong to arbitrary open bases of the spaces $Y$ and $X$, respectively. In particular, for metric spaces this definition of continuity can be transferred into the usual definition familiar from courses of mathematical analysis. If $f:X\to Y$ is continuous at each point $x\in X$, then it is called a continuous mapping from $X$ into $Y$. Each of the following conditions is necessary and sufficient for continuity of $f:X\to Y$. a) If $x$ is a proximate point of a set $M\subset X$, then $f(x)$ is a proximate point of the set $f(M)$ in $Y$. b) The complete pre-image $f^{-1}(\Gamma)$ of any open set $\Gamma$ in any $Y$ is open in $X$. There are analogues for closed sets.

Given a (continuous) mapping $f$ from a topological space $X$ into a topological space $Y$ and a subspace $X_0$ of $X$, the mapping $f$ maps $X_0$ into $Y$ and is continuous (it is called the restriction of $f:X\to Y$ to the subspace $X_0$).

An important type of continuous mappings are the so-called quotient mappings (cf. Quotient mapping). They are characterized by the following condition. A set $B\subset Y$ is open (closed) in $Y$ if and only if $f^{-1}(B)$ has the same property in $X$. If a continuous mapping $f$ from $X$ onto $Y$ is bijective, then the inverse mapping $f^{-1}:Y\to X$ is defined, but it need not be continuous. If it is continuous, then each of the mappings $f,f^{-1}$ maps the topologies of $X,Y$ bijectively onto each other. Each of the two bijective mappings $f$ and $f^{-1}$ is called a topological mapping, or a homeomorphism. Two spaces $X$ and $Y$ that can be homeomorphically mapped onto each other are said to be homeomorphic, or topologically equivalent.

A continuous mapping $f:X\to Y$ is said to be irreducible if no closed set $M$, other than $X$, is mapped onto the whole of $Y$.

The concrete study of topological spaces consists primarily of the separation from the general class of those spaces of subclasses characterized by some conditions or axioms additional to those defining topologies. These additional axioms can be of various kinds. First of all there is a group of so-called separation axioms. The first separation axiom was the Hausdorff axiom, requiring that any two distinct points of the space can be separated by means of neighbourhoods, i.e. are contained in disjoint open sets (two or more sets are disjoint if no two of them have common elements). Hausdorff's separation axiom is also called the $T_2$-axiom, and a topological space satisfying it is called a Hausdorff space or a $T_2$-space. Having defined these spaces, F. Hausdorff in 1914 for the first time discovered sufficiently broad and at the same time sufficiently rich properties of a class of topological spaces, thereby fulfilling a need in mathematics that was very urgent at that time. The subsequent development of general topology emerges from precisely the Hausdorff spaces. A weakening of the $T_2$-axiom is the $T_1$-axiom: Each of two points of a topological space has a neighbourhood not containing the other point. This requirement is equivalent to the requirement that any finite subset is closed. A space satisfying this requirement is called a $T_1$-space. A still broader class of topological spaces is formed by the $T_0$-spaces, i.e. spaces in which the $T_0$-axiom (the Kolmogorov axiom) is satisfied: Given two distinct points, then at least one has a neighbourhood not containing the other point. $T_0$-spaces may consist of a finite number or even only of two points, if one point forms a closed but not an open set while the other, conversely, forms an open but not a closed set (a connected digon).

The $T_3$-axiom requires that an arbitrary point of the space and an arbitrary closed set not containing this point can be separated by neighbourhoods, i.e. are contained, respectively, in two disjoint open sets. A space satisfying the $T_3$-axiom is called a $T_3$-space. A $T_3$-space need not satisfy the $T_1$-axiom. (Example: the connected digon.) $T_3$-spaces satisfying the $T_1$-axiom are said to be regular (cf. Regular space); they are Hausdorff spaces. A topological space is called a $T_4$-space if every two disjoint closed sets in it have disjoint neighbourhoods. $T_4$-spaces that are simultaneously $T_1$-spaces are called normal (cf. Normal space); they are regular and Hausdorff. Any subspace of a space satisfying one of the axioms $T_i$, $i=0,1,2,3$, also satisfies that axiom $T_i$, i.e. the axioms $T_i$ are inherited by all subspaces of the given space. The axiom $T_4$ does not possess this property: There exist normal spaces $X$ not all subspaces (even not all open subspaces) of which are normal. However, if $X_0$ is a closed set in a normal space $X$, then the subspace $X_0$ is normal.

Until now the separation of points and sets has been understood in the sense of the presence of disjoint neighbourhoods. However, in modern topology so-called functional separation, originally introduced by P.S. Urysohn in 1924, is also important. Two sets $A$ and $B$ in a topological space are called functionally separable if there exists a real-valued, continuous and bounded function $f$ on the whole space $X$ taking the value $0$ at all points of $A$ and the value $1$ at all points of $B$. In a normal space any two disjoint closed sets are functionally separable (Urysohn's lemma). Conversely, functional separation of two arbitrary sets implies their separation by neighbourhoods. In particular, functional separation of a point and a set implies their separation by neighbourhoods in the given space. However, if the space is regular, hence every point and every closed set not containing it have disjoint neighbourhoods, it does not follow that every point and set are functionally separable. Thus, the following property of complete regularity is stronger than the property of regularity. A space is called completely regular if any point and any closed set not containing it are functionally separable (cf. Completely-regular space). Among the (completely-regular) spaces satisfying this condition the most important are the completely-regular $T_1$-spaces, also called Tikhonov spaces (cf. Tikhonov space). In particular, the underlying space of any $T_0$-topological group is completely regular but need not be normal.

As well as the separation axioms, the so-called conditions of compactness type are significant for the theory of topological spaces. They are based on the consideration of (open) coverings. A family $\Sigma$ of (open) sets of a given topological space $X$ is called a covering of $X$ (cf. Covering (of a set)) if every point $x\in X$ is contained in at least one set that is an element of $\Sigma$. If every element of a covering $\alpha'$ is a subset of at least one element of a covering $\alpha$, then $\alpha'$ is said to refine $\alpha$, or $\alpha'$ is said to be finer than $\alpha$, or $\alpha'$ is said to follow $\alpha$ in the partially ordered set of coverings of the given space. A particular case of $\alpha'$ refining $\alpha$ is the case when $\alpha'$ is contained in $\alpha$ (i.e. each element of $\alpha'$ is also an element of $\alpha$). Some conditions of compactness type assume that two classes of (open) coverings of a space $X$ have been given: a class $\mathfrak{A}$ and a class $\mathfrak{B}$ such that $\mathfrak{B}\subset\mathfrak{A}$, i.e. each covering in $\mathfrak{B}$ is also in $\mathfrak{A}$. The condition of $(\mathfrak{A},\mathfrak{B})$-compactness requires that for each covering in $\mathfrak{A}$ there is covering in $\mathfrak{B}$ refining it. Among all conditions of this type the most important one is obtained if $\mathfrak{A}$ is the class of all open coverings of a space and $\mathfrak{B}$ is the subclass of finite coverings, i.e. coverings consisting of a finite number of elements. This condition is called the compactness condition; it is equivalent to the so-called Borel–Lebesgue condition: For each open covering $\alpha$ of a space $X$ there is a finite covering $\alpha'$ of $X$ contained in $\alpha$. Hausdorff spaces satisfying the compactness condition are called Hausdorff compacta. They are all normal. Metrizable Hausdorff compacta (Hausdorff compacta with a countable base) are called compacta. Nowadays another terminology prevails, in which Hausdorff compacta are called compact Hausdorff topological spaces, and the metrizable case is terminologically not distinguished (cf. also Compact space).

If as $\mathfrak{A}$ one takes the class of countable open coverings, while still taking for $\mathfrak{B}$ the subclass of finite coverings, then the corresponding $(\mathfrak{A},\mathfrak{B})$-compactness is called countable compactness (cf. Compact set, countably). For metrizable spaces, and also for Hausdorff spaces with a countable base, the conditions of being compact and of being countably compact are equivalent. If as $\mathfrak{A}$ one takes the class of all open coverings and as $\mathfrak{B}$ the class of countable coverings, then one obtains the condition of final compactness. In the formulation of this condition one may require (as in the case of compactness) that the covering $\alpha'\in\mathfrak{B}$ is not only a refinement of $\alpha$, but is contained in it.

An important class of spaces, called locally compact spaces (cf. Locally compact space), is defined by the requirement that every point $x$ of the given space $X$ has a neighbourhood $O_x$ whose closure in $X$ is a compact subspace. Any locally compact Hausdorff space, and only such, can be considered as an open set of a certain Hausdorff compactum $X$, moreover, $X$ is obtained from $X$ by adjunction of only one point, $x$, and the topology of $X$ is unambiguously determined by this last requirement and by the topology of $X$; the Hausdorff compactum $X$ is said to be the one-point compactification or Aleksandrov compactification of $X$.

Next to conditions of compactness the most important condition of compactness type is the condition of paracompactness (cf. Paracompactness criteria), requiring that every open covering $\alpha$ of a given space $X$ can be refined by a locally finite open covering $\alpha'$ (a family of sets of a topological space is said to be locally finite in it if each point has a neighbourhood that intersects with only a finite number of sets of the family). Here one cannot require that $\alpha'$ is contained in $\alpha$. All metrizable spaces are paracompact Hausdorff spaces.

#### References

 [1] P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian) [2] N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French) [3] K. Kuratowski, "Topology" , 1 , PWN & Acad. Press (1966) (Translated from French) [4] P.S. Aleksandrov, B.A. Pasynkov, "Introduction to dimension theory" , Moscow (1973) (In Russian)

Among the equivalent ways of determining a topology on a space, convergence structures should be noted. (All of the usual approaches — open sets, closure operations, convergence structures — also give rise to further generalizations.) The early descriptions of convergence used external devices such as directed sets (cf. [a3] and Directed set), but there is an intrinsic version. It is technically intrinsic in that the apparatus is determined by the underlying set $X$; psychologically it is far from intrinsic, introducing cardinal numbers $\mathfrak{m}$. Then for every power set $X^{\mathfrak{m}}$, for every ultrafilter $\mathcal{F}$ on the index set $\mathfrak{m}$, one must specify which points $x_0$ are limits along $\mathcal{F}$ of which $\mathfrak{m}$-tuples $\left\{ x_\alpha \right\}\in X^{\mathfrak{m}}$. The axioms for the compact Hausdorff case, where each $\mathfrak{m}$-tuple has a unique limit along each ultrafilter, were formulated in [a2]; for the general case, in [a1]. In applications, convergence with auxiliary devices, especially convergence of generalized sequences (nets, cf. Generalized sequence), is much more often used.