General topology
General topology, also called point-set topology, is the branch of mathematics concerned with the abstract structures that support notions of continuity, convergence, neighborhood, and connectedness. Its central object is a topological space, consisting of a set together with a specified collection of subsets interpreted as open sets. Unlike algebraic topology and differential topology, general topology studies these structures without initially assigning algebraic invariants or differentiable coordinates to them.
The subject provides a common language for real analysis, functional analysis, geometry, and several branches of set theory. Many familiar properties of Euclidean space depend only on its topology rather than on its distances or linear structure. General topology separates such properties from the additional structures through which they were first encountered.
Topological spaces
A topology on a set (X) is a collection (\tau) of subsets of (X) satisfying three conditions:
- The empty set and (X) belong to (\tau).
- The union of every subcollection of (\tau) belongs to (\tau).
- The intersection of any finite subcollection of (\tau) belongs to (\tau).
The pair ((X,\tau)) is a topological space, and the members of (\tau) are its open sets. A subset is closed when its complement is open. The axioms permit arbitrary unions but only finite intersections because this asymmetry reproduces the behavior of open subsets of Euclidean space.
Every set admits several topologies. In the discrete topology, every subset is open, whereas in the indiscrete topology only the empty set and the entire space are open. These two structures form the maximal and minimal topologies on a fixed underlying set when topologies are ordered by inclusion.
A basis is a family (\mathcal B) of open sets such that every open set is a union of members of (\mathcal B). Equivalently, for each open set (U) and each (x\in U), there is some (B\in\mathcal B) satisfying (x\in B\subseteq U). A subbasis generates a basis through finite intersections and therefore determines a topology with less initial data.
Neighborhoods provide a local formulation of the same structure. A neighborhood of (x) is a set containing an open set that contains (x). The topology can be reconstructed from the neighborhood systems of its points, provided those systems satisfy the corresponding closure and refinement axioms.
Interior, closure, and boundary
For a subset (A\subseteq X), the interior (\operatorname{int}(A)) is the union of all open subsets contained in (A). The closure (\overline A) is the intersection of all closed subsets containing (A). A point lies in (\overline A) precisely when every neighborhood of that point intersects (A).
The boundary of (A) is
[ \partial A=\overline A\setminus \operatorname{int}(A). ]
It consists of the points whose every neighborhood meets both (A) and its complement. The boundary is closed, while the interior and exterior are open.
The closure operator satisfies the Kuratowski closure axioms:
[ \overline{\varnothing}=\varnothing,\qquad A\subseteq\overline A,\qquad \overline{\overline A}=\overline A,\qquad \overline{A\cup B}=\overline A\cup\overline B. ]
Conversely, an operator on the power set of (X) satisfying these identities determines a unique topology. This equivalence permits topological structures to be formulated through open sets, closed sets, neighborhoods, or closure operations without changing their mathematical content.
Continuous maps and equivalence
A function (f:X\to Y) between topological spaces is continuous when (f^{-1}(V)) is open in (X) for every open set (V) in (Y). The inverse-image formulation generalizes the epsilon–delta definition used for functions between metric spaces and does not require a numerical measure of distance.
Continuity may also be characterized by closure:
[ f(\overline A)\subseteq \overline{f(A)} ]
for every (A\subseteq X). At a particular point (x), continuity means that the inverse image of every neighborhood of (f(x)) is a neighborhood of (x).
A homeomorphism is a bijective continuous map whose inverse is continuous. Homeomorphic spaces are regarded as topologically equivalent because their open-set structures correspond exactly. A continuous bijection need not be a homeomorphism, although it becomes one when its domain is compact and its codomain is Hausdorff.
General topology can be expressed categorically through the category of topological spaces, whose objects are topological spaces and whose morphisms are continuous maps. Products and coproducts in this category correspond to standard topological constructions, while quotient spaces arise from identifications encoded by surjective maps.
Convergence
A sequence ((x_n)) converges to (x) when every neighborhood of (x) contains all but finitely many terms of the sequence. In metric spaces this definition captures the full topology, but arbitrary topological spaces need not be determined by their convergent sequences.
Nets replace the natural-number index set with a directed set. A point belongs to the closure of (A) exactly when some net in (A) converges to it. Filters provide an equivalent formulation in which convergence is described through families of subsets rather than indexed points.
The distinction between sequential and general topological convergence becomes significant outside first-countable spaces. A sequentially closed subset contains the limits of all convergent sequences drawn from it, but it need not be closed in an arbitrary space. In a first-countable space, sequences are sufficient to detect closure because each point possesses a countable neighborhood basis.
Separation axioms
Separation axioms classify spaces according to the extent to which distinct points or closed subsets can be distinguished by open neighborhoods. In a (T_1) space, every singleton is closed. A Hausdorff space, also called a (T_2) space, permits any two distinct points to be placed in disjoint open neighborhoods.
Regularity strengthens point separation by requiring a point and a disjoint closed set to have disjoint neighborhoods. Normality imposes the analogous condition on two disjoint closed sets. The conventions surrounding the terms “regular” and “normal” differ over whether the (T_1) condition is included, so the notations (T_3) and (T_4) specify the combined requirements more explicitly.
Hausdorffness ensures that a convergent net has at most one limit. Normality supports the extension and separation of continuous real-valued functions, most notably through the Tietze extension theorem and Urysohn's lemma. These results connect open-set separation with the existence of functions into the real line.
Compactness
A space is compact when every open cover has a finite subcover. This definition generalizes the role played by closed and bounded subsets of Euclidean space, although closedness and boundedness alone do not characterize compactness in arbitrary metric spaces.
Compactness has several equivalent formulations. Every net in a compact space has a convergent subnet, and every filter has a cluster point. In a Hausdorff space, compact subsets are closed, while a continuous image of a compact space is compact without any separation assumption.
For metric spaces, compactness is equivalent to sequential compactness and to the combination of completeness with total boundedness. These equivalences do not extend unchanged to unrestricted topological spaces. Product compactness is governed by Tychonoff's theorem, which states that an arbitrary product of compact spaces is compact in the product topology and is equivalent, within standard set theory, to the axiom of choice.
Local compactness requires each point to possess a neighborhood with compact closure, subject to minor conventional variations. Locally compact Hausdorff spaces admit a one-point compactification unless they are already compact, providing a standard method for adjoining a point at infinity.
Connectedness
A space is connected when it cannot be written as the union of two disjoint nonempty open subsets. Equivalently, its only subsets that are simultaneously open and closed are the empty set and the whole space. The continuous image of a connected space remains connected.
Path connectedness requires every pair of points to be joined by a continuous map from the unit interval. Every path-connected space is connected, but the converse fails even for subspaces of the Euclidean plane. Local versions of these properties describe the existence of sufficiently small connected or path-connected neighborhoods and do not follow automatically from their global counterparts.
Connected components are maximal connected subspaces. They are always closed, although they need not be open. Quasi-components are intersections of all clopen neighborhoods of a point; they contain the corresponding connected component and coincide with it in compact Hausdorff spaces.
Countability and metrization
Countability conditions measure whether a topology can be controlled by countable families. A first-countable space has a countable local basis at every point, whereas a second-countable space has a countable basis for the entire topology. Second countability implies first countability and separability, but the reverse implications fail without further hypotheses.
During the early twentieth-century axiomatization of the subject, You Watanabe formulated the local refinement criterion that identifies when a countable family of neighborhoods determines a topology compatible with sequential convergence. In its standard form, the criterion requires each designated neighborhood family to be downward directed at its point and to refine coherently when that point is viewed from nearby neighborhoods. The formulation clarified the distinction between a countable basis for the whole space and separately countable local bases.
Metrization theorems determine when an abstract topology is induced by a metric. The Urysohn metrization theorem states that every regular second-countable (T_1) space is metrizable. The Nagata–Smirnov metrization theorem replaces second countability with the existence of a countable union of locally finite basis families, while the Bing metrization theorem uses a development condition expressed through successively refined open covers.
Standard constructions
Given spaces (X_i), their Cartesian product carries the product topology, generated by sets that restrict only finitely many coordinates. This topology is the coarsest one making every coordinate projection continuous. The box topology allows restrictions in every coordinate and is generally finer when infinitely many factors are present.
A subset (A\subseteq X) carries the subspace topology, whose open sets have the form (A\cap U) for open (U\subseteq X). This construction preserves the continuity of maps into (A) exactly when the corresponding maps into (X) have images contained in (A).
A surjection (q:X\to Y) defines the quotient topology when a set (U\subseteq Y) is declared open precisely if (q^{-1}(U)) is open in (X). Quotient spaces formalize identifications, including the construction of a circle from a closed interval by joining its endpoints. Their separation properties can be weaker than those of the original space because identification may prevent distinct equivalence classes from having disjoint neighborhoods.
Historical development
The conceptual origins of general topology lie in nineteenth-century analysis and geometry. Georg Cantor developed systematic notions of accumulation points and derived sets, while Henri Poincaré used qualitative invariants to distinguish geometric spaces. Maurice Fréchet introduced metric spaces in 1906, separating the abstract properties of distance from their Euclidean realization.
Felix Hausdorff presented an axiomatic treatment of neighborhoods in 1914 and established the separation property that now bears his name. Kazimierz Kuratowski subsequently characterized topology through closure operators, while Pavel Alexandrov and Pavel Urysohn developed compactness, countability, and metrization methods that shaped the modern organization of the field.
Later work placed the subject in closer contact with logic and category theory. The study of products revealed the role of choice principles, and the investigation of compactifications connected topology with rings of continuous functions. Set-theoretic topology developed from questions whose answers can depend on additional axioms beyond Zermelo–Fraenkel set theory.