General topology
General topology, also known as point-set topology, is a branch of mathematics concerned with the study of topological spaces, which are sets equipped with a structure called a topology consisting of a collection of subsets known as open sets that satisfy specific axioms: the empty set and the whole space are open, arbitrary unions of open sets are open, and finite intersections of open sets are open.[1] This framework generalizes the notions of continuity, limits, and neighborhoods from metric spaces to more abstract settings, allowing the analysis of properties preserved under continuous deformations.[2] Unlike algebraic topology, which uses algebraic tools like group theory to study global properties such as homotopy groups, general topology relies on set-theoretic methods to examine local properties and separation axioms.[2] The development of general topology emerged in the early 20th century as a unification of ideas from analysis, geometry, and set theory, with foundational contributions from Felix Hausdorff, who formalized the concept of topological spaces in his 1914 book Grundzüge der Mengenlehre.[3] Earlier precursors include 19th-century work on continuous functions and compactness, such as by Karl Weierstrass and the Heine-Borel theorem attributed to Émile Borel. The axiomatic approach was further developed in the 1920s, with key contributions from mathematicians like Maurice Fréchet and Pavel Alexandrov.[3] By the mid-20th century, general topology had become a cornerstone of modern mathematics, influencing fields from functional analysis to theoretical computer science.[4] Central concepts in general topology include continuous functions, which preserve the topological structure by mapping open sets to open sets; compactness, a property ensuring every open cover has a finite subcover, generalizing boundedness in Euclidean spaces; and connectedness, which describes spaces that cannot be divided into disjoint nonempty open subsets.[5] Separation axioms, such as T1 (points are closed) and Hausdorff (T2, distinct points have disjoint neighborhoods), classify topological spaces by their ability to distinguish points.[6] Bases and subbases provide ways to generate topologies, while closure and interior operators extend the structure to closed sets and other derived notions. These tools enable the study of convergence, completeness, and metrizability, with applications in understanding the real line, function spaces, and abstract geometric objects.[7]History
Early development
The foundations of general topology trace back to late 19th-century advancements in analysis and geometry, where mathematicians began addressing properties invariant under continuous transformations. Bernhard Riemann's work on Riemann surfaces, introduced in his 1851 doctoral dissertation, provided a geometric model for multi-valued functions in complex analysis, incorporating notions of connectivity, branching, and genus that highlighted qualitative features beyond metric considerations. These ideas influenced the study of surfaces as abstract objects, foreshadowing topological invariants.[8] Parallel developments in set theory by Georg Cantor during the 1870s and 1890s contributed essential tools for point-set topology. In his 1872 paper published in Mathematische Annalen, Cantor defined the derived set as the collection of limit points of a given set and characterized closed subsets of the real line as those containing all their limit points, laying the groundwork for concepts of accumulation, isolation, and continuity in terms of sets rather than functions.[9] Cantor's subsequent explorations of perfect sets and the Cantor set in the 1880s further emphasized dense, nowhere-dense structures, bridging analysis with emerging topological ideas. A significant intuitive leap occurred in 1895 with Henri Poincaré's seminal paper "Analysis Situs," published in the Journal de l'École Polytechnique, which focused on the "analysis of position" or qualitative geometry of manifolds. Poincaré introduced homology groups and Betti numbers to classify surfaces and curves up to homeomorphism, emphasizing invariants like connectivity and orientability that resist deformation, thus motivating topology as a distinct discipline from metric geometry.[10] This work influenced the shift toward abstract, deformation-invariant properties. In 1906, Maurice Fréchet advanced abstraction in his thesis "Sur quelques points du calcul fonctionnel," published in Rendiconti del Circolo Matematico di Palermo, by defining metric spaces as sets equipped with a distance function satisfying basic axioms, independent of any embedding in Euclidean space.[11] This framework unified prior work on function spaces and convergence, serving as a direct precursor to non-metric topologies by generalizing notions of nearness. Felix Hausdorff's 1914 monograph Grundzüge der Mengenlehre synthesized these threads into a rigorous axiomatic system. Hausdorff formalized topological spaces using neighborhood filters with four axioms—symmetry, inclusion, minimality, and intersection—while introducing separation axioms like T1 and T2 to distinguish points, enabling the study of abstract spaces without reliance on metrics or order.[12] This text established general topology as a foundational branch of mathematics, paving the way for axiomatic refinements.Key contributors and milestones
In the 1920s, the Polish school of mathematics, centered in Warsaw and Lwów, advanced general topology through axiomatic approaches, with Kazimierz Kuratowski playing a pivotal role by introducing the closure operator axioms in 1922, utilizing Boolean algebra to define topological structures independently of points.[13] This work complemented earlier efforts and solidified the school's emphasis on point-set topology, as seen in foundational publications like Fundamenta Mathematicae starting in 1920.[13] Concurrently, the Soviet school, led by Pavel Alexandrov and Pavel Urysohn, developed the neighborhood system axioms for topological spaces in their 1924 collaboration, providing an intrinsic characterization of continuity and openness without reliance on metrics.[14] Their joint paper "Zur Theorie der topologischen Räume" formalized these ideas, establishing a framework that influenced subsequent axiomatizations.[14] A landmark achievement was Urysohn's metrization theorem, announced in 1924 and published posthumously in 1925, stating that a second-countable, regular Hausdorff space is metrizable.[14] The proof outline involves first applying Urysohn's lemma to construct continuous functions separating points from closed sets in the normal space, then embedding the space into a product of intervals via these functions to yield a countable basis, and finally deriving a metric from the embedding.[14] In 1930, Andrey Tychonoff proved that the product of any collection of compact topological spaces is compact in the product topology, a result that extended finite-product compactness to arbitrary families and relied on the axiom of choice. This theorem, detailed in his paper "Über die topologische Erweiterung von Räumen," became foundational for infinite-dimensional topology. During the 1930s, Andrey Kolmogorov contributed to the structure of Borel sets by investigating the hierarchy of Borel classes and posing problems on its length, as in his 1935 query in Fundamenta Mathematicae.[15] Roman Sikorski advanced related work on Boolean algebras representing Borel structures, linking set-theoretic operations to topological measurability in Polish topological contexts.[16] Post-World War II developments included André Weil's 1937 introduction of uniform structures, providing a generalization beyond metrics for notions like completeness and uniform continuity. Richard Arens' 1946 work developed topologies for spaces of transformations using these uniform ideas, enabling the study of convergence in function spaces.[17][18] In the 1940s, Norman Steenrod's work on fiber bundles, culminating in his 1951 monograph The Topology of Fibre Bundles, provided axiomatic foundations that bridged general and algebraic topology, influencing classifications and cohomology applications.Topological spaces
Definition of a topology
In general topology, a topological space is formally defined as a pair (X, \tau), where X is a set and \tau is a collection of subsets of X satisfying certain axioms; this axiomatic framework, equivalent to the closure operator axioms introduced by Kazimierz Kuratowski in 1922, provides the foundation for abstracting notions of continuity and proximity without relying on a specific metric.[19] The collection \tau is called a topology on X, and its elements are termed open sets. This definition assumes familiarity with basic set-theoretic operations, such as unions and intersections of families of sets.[12] The axioms for \tau to be a topology on X are as follows:- The empty set \emptyset and the whole set X belong to \tau.
- The union of any (possibly infinite) collection of sets in \tau is again in \tau.
- The intersection of any finite collection of sets in \tau is again in \tau.
Bases and subbases
A base (or basis) for a topology \tau on a set X is a subcollection \mathcal{B} \subseteq \tau of open sets such that every open set in \tau can be expressed as an arbitrary union of elements from \mathcal{B}.[20] Moreover, the union of all sets in \mathcal{B} must equal X.[5] To determine whether a given collection \mathcal{B} of subsets of X forms a basis for some topology on X, it must satisfy two conditions: first, the union of elements in \mathcal{B} covers X; second, for any two elements B_1, B_2 \in \mathcal{B} and any point x \in B_1 \cap B_2, there exists an element B_3 \in \mathcal{B} such that x \in B_3 \subseteq B_1 \cap B_2.[20] If \mathcal{B} meets these criteria, the topology generated by \mathcal{B} consists of all arbitrary unions of elements from \mathcal{B}, and \mathcal{B} serves as a basis for this topology.[5] Equivalently, for a collection \mathcal{B} to be a basis for an existing topology \tau, it must hold that for every open set U \in \tau and every x \in U, there exists B \in \mathcal{B} with x \in B \subseteq U.[5] A subbasis (or subbase) for a topology on X is a collection \mathcal{S} of subsets of X whose union covers X.[20] The topology \sigma(\mathcal{S}) generated by \mathcal{S} is formed by first taking all finite intersections of elements from \mathcal{S} (which yields a basis), and then taking all arbitrary unions of those finite intersections.[5] Formally, \sigma(\mathcal{S}) = \left\{ \bigcup_{i \in I} \left( \bigcap_{j=1}^{n_i} S_{i j} \right) \;\middle|\; I \text{ any index set}, \; n_i \in \mathbb{N}, \; S_{i j} \in \mathcal{S} \right\}, where the empty union is the empty set and the full space X arises from appropriate choices.[20] In the standard topology on the Euclidean space \mathbb{R}^n, the collection of all open balls forms a basis.[20] For a simple illustration, consider X = \{a, b, c, d\} with subbasis \mathcal{S} = \{\{a, b, c\}, \{b, c, d\}\}; the generated topology is \{\emptyset, \{b, c\}, \{a, b, c\}, \{b, c, d\}, X\}.[20]Subspaces and quotient topologies
In general topology, the subspace topology provides a natural way to endow a subset of a topological space with its own topology, inheriting the structure from the ambient space. Let (X, \tau) be a topological space and Y \subseteq X a subset. The subspace topology \tau_Y on Y is defined as \tau_Y = \{ U \cap Y \mid U \in \tau \}.[21] This construction ensures that the inclusion map i: Y \hookrightarrow X is continuous, and open (respectively, closed) sets in the subspace are precisely the intersections of open (closed) sets in X with Y.[2] Thus, subspaces inherit key topological properties such as openness and closedness relative to the ambient space, allowing for the study of induced structures on subsets without altering the original topology.[22] The subspace topology also interacts naturally with bases of the original topology: if \mathcal{B} is a base for \tau, then \{ B \cap Y \mid B \in \mathcal{B} \} forms a base for \tau_Y.[20] In contrast, the quotient topology constructs a topology on a set by identifying points via an equivalence relation or surjective map, often used to model gluing or collapsing in spaces. Let q: X \to Y be a surjective map from a topological space (X, \tau_X) to a set Y. The quotient topology \tau_Y on Y is the finest topology such that q is continuous, defined by \tau_Y = \{ V \subseteq Y \mid q^{-1}(V) \in \tau_X \}.[23] Equivalently, for an equivalence relation \sim on X, the quotient space X / \sim carries the topology where a set U \subseteq X / \sim is open if the preimage under the canonical projection \pi: X \to X / \sim is open in X.[24] This topology ensures that saturated open sets in X (those unions of equivalence classes) map to open sets in Y, and similarly for closed sets, preserving relevant openness and closedness properties under the identification.[22] A classic example is the real projective line \mathbb{RP}^1, obtained as the quotient of the circle S^1 under the antipodal identification z \sim -z, where opposite points are glued together; this space is homeomorphic to S^1 itself but illustrates the quotient construction without relying on coordinate charts.[25]Examples of topological spaces
Discrete and indiscrete topologies
The discrete topology on a nonempty set X is defined as the collection of all subsets of X as open sets, making it the power set \mathcal{P}(X).[26] This topology is the finest (largest) possible on X, containing every open set from any other topology on the same set. In the discrete topology, every singleton \{x\} for x \in X is open, and consequently, every subset of X is both open and closed.[26] Any function f: (X, \tau_d) \to (Y, \tau_Y), where \tau_d is the discrete topology on X and \tau_Y is any topology on Y, is continuous, since the preimage f^{-1}(V) of any open V \in \tau_Y is a subset of X, hence open in \tau_d. The discrete topology satisfies the Hausdorff separation axiom, as distinct points x, y \in X can be separated by the open singletons \{x\} and \{y\}. Moreover, a discrete space is compact if and only if X is finite, because an infinite discrete space admits an open cover by singletons with no finite subcover. The indiscrete topology (also called the trivial topology) on a set X consists solely of the empty set \emptyset and X itself as open sets, making it the coarsest (smallest) topology on X.[27] In this topology, the only closed sets are also \emptyset and X, so no proper nonempty subset, including singletons, is closed unless |X| \leq 1. Any function f: (Y, \tau_Y) \to (X, \tau_i), where \tau_i is the indiscrete topology on X and \tau_Y is any topology on Y, is continuous, because the preimage f^{-1}(U) of any open U \in \tau_i (either \emptyset or X) is either \emptyset or Y, both open in \tau_Y. However, a function f: (X, \tau_i) \to (Y, \tau_Y) is continuous only if it is constant when Y is nontrivial (e.g., with at least two points separable by open sets), as nonconstant maps would map some open V \in \tau_Y to a preimage that is neither \emptyset nor X.[28] The discrete topology arises naturally on finite sets in many mathematical contexts, such as when considering sets without additional structure, ensuring all subsets are distinguishable topologically.[29] The indiscrete topology models trivial situations where no proper distinctions are made, such as the whole space in certain quotient constructions or as a baseline for comparing coarser topologies.Cofinite and cocountable topologies
The cofinite topology on a set X is the topology whose open sets consist of the empty set and all subsets of X whose complements are finite.[30] Equivalently, the closed sets in this topology are the finite subsets of X and X itself.[31] This topology coincides with the discrete topology when X is finite.[32] For an infinite set X equipped with the cofinite topology, the space is T_1, as singletons are finite and hence closed.[4] However, it is not Hausdorff, since any two nonempty open sets have nonempty intersection: their complements are finite, so their union is finite, and thus the intersection is cofinite and nonempty given that X is infinite.[4] The space is compact, as any open cover admits a finite subcover: select one open set U from the cover, whose complement is finite; the remaining points in the complement can then be covered by finitely many additional open sets from the cover.[33] It is also connected: if X = U \cup V with U and V nonempty, disjoint, and open, then the complements X \setminus U = V and X \setminus V = U are both finite, implying X is finite, a contradiction.[34] When X is uncountable with the cofinite topology, the space is not second-countable: a countable basis \mathcal{B} would yield a countable union of the finite complements X \setminus B for B \in \mathcal{B}, leaving points outside this union uncovered by any basis element in a way that generates all opens.[35] If X is countable and infinite, such as the natural numbers, the cofinite topology is not Hausdorff, and sequences exhibit nonunique convergence; for instance, a sequence of distinct natural numbers converges to every point in the space, as any neighborhood of a limit point excludes only finitely many elements and thus contains infinitely many terms.[36] The cocountable topology is defined on an uncountable set X as the topology whose open sets are the empty set and all subsets whose complements are countable.[37] Equivalently, the closed sets are the countable subsets of X and X itself.[38] On a countable set, this reduces to the discrete topology.[6] For uncountable X with the cocountable topology, the space is T_1, since singletons are countable and hence closed.[39] It is not Hausdorff, as any two nonempty open sets intersect: their complements are countable, so their union is countable, leaving the intersection uncountable and nonempty.[39] The space is connected—in fact, hyperconnected—because every pair of nonempty open sets has nonempty intersection.[39] Unlike the cofinite case, it is not compact: consider a countable infinite subset \{y_n\}_{n=1}^\infty \subset X; the open sets V_n = X \setminus \{y_1, \dots, y_n\} form a countable open cover of X with no finite subcover, as any finite union V_1 \cup \cdots \cup V_m = V_m misses y_{m+1}.[40] It is also not second-countable when X is uncountable, by an argument analogous to the cofinite case but involving countable rather than finite complements.[32]Topologies on real and complex numbers
The standard topology on the real numbers \mathbb{R} is the order topology induced by the usual linear order \leq, where a subbasis consists of the open rays (-\infty, a) and (a, \infty) for all a \in \mathbb{R}. A basis for this topology is given by the open intervals (a, b) = \{x \in \mathbb{R} \mid a < x < b\} for a < b. This topology makes \mathbb{R} a connected space, as it cannot be expressed as a union of two nonempty disjoint open sets, but \mathbb{R} is not compact, since the open cover \{(n, n+2) \mid n \in \mathbb{Z}\} has no finite subcover.[41][42][43] Another notable topology on \mathbb{R} is the lower limit topology, also called the Sorgenfrey topology, generated by the basis of half-open intervals [a, b) = \{x \in \mathbb{R} \mid a \leq x < b\} for a < b. This topology is finer than the standard topology, meaning every standard open set is Sorgenfrey-open, but it includes additional open sets. The Sorgenfrey line is hereditarily Lindelöf, as every subspace has the Lindelöf property (every open cover has a countable subcover), yet it is not second-countable, since any countable collection of basis elements cannot cover all singletons in an uncountable disjoint family.[44][45] The complex numbers \mathbb{C} carry the standard topology by identifying \mathbb{C} with \mathbb{R}^2 via the map z = x + iy \mapsto (x, y), endowing it with the product topology (or equivalently, the Euclidean topology on \mathbb{R}^2). Open sets are unions of open disks B_r(\zeta) = \{z \in \mathbb{C} \mid |z - \zeta| < r\} for \zeta \in \mathbb{C} and r > 0. Like \mathbb{R}, \mathbb{C} is connected but not compact in this topology.[46][47] A non-Hausdorff topology on \mathbb{C}, viewed as the affine line \mathbb{A}^1(\mathbb{C}), is the Zariski topology, where closed sets are the whole space \mathbb{C} or finite subsets (zeros of nonzero polynomials in \mathbb{C}). Since any two nonempty open sets (complements of finite sets) intersect, the space fails to separate distinct points with disjoint open neighborhoods.[48][49] As a non-Hausdorff variant on a line-like space, consider the double-pointed line (or line with two origins), constructed by taking two copies of \mathbb{R} and identifying all points except the origins, resulting in a space homeomorphic to \mathbb{R} away from the two distinct origin points o_1 and o_2. Basic open sets around points other than the origins are standard intervals, but neighborhoods of o_1 and o_2 cannot be disjoint while containing each, violating the Hausdorff axiom, though the space is otherwise locally Euclidean.[50][51]Continuous functions
Primary definitions
In general topology, a function f: X \to Y between topological spaces (X, \mathcal{T}_X) and (Y, \mathcal{T}_Y) is defined to be continuous if the preimage f^{-1}(V) of every open set V \in \mathcal{T}_Y is an open set in X.[52] This open-set definition generalizes the intuitive notion of continuity from metric spaces, where it corresponds to the \epsilon-\delta condition, but applies to arbitrary topological spaces without relying on distances.[1] An equivalent characterization of continuity is that the preimage f^{-1}(C) of every closed set C in Y is closed in X.[52] Another equivalent condition is that the graph of f, defined as \Gamma_f = \{(x, f(x)) \mid x \in X\} \subseteq X \times Y equipped with the product topology, is a closed subset of X \times Y.[1] Related to continuity are the concepts of open and closed maps. A function f: X \to Y is an open map if the image f(U) of every open set U \in \mathcal{T}_X is open in Y, and it is a closed map if the image f(C) of every closed set C in X is closed in Y.[53] While continuous functions need not be open or closed, these properties highlight how f interacts with the topological structures of X and Y. Continuous functions preserve certain closure properties: for any subset A \subseteq X, the image of the closure satisfies f(\overline{A}) \subseteq \overline{f(A)}, where \overline{A} denotes the closure of A in X and \overline{f(A)} the closure of f(A) in Y.[1] This inclusion reflects how continuity ensures that limits and accumulations in the domain map into accumulations in the codomain.Alternative characterizations
In general topological spaces, continuity of a function f: X \to Y at a point x \in X can be equivalently characterized using neighborhoods: for every neighborhood V of f(x) in Y, there exists a neighborhood U of x in X such that f(U) \subseteq V.[54] This formulation emphasizes local preservation of nearness and generalizes the epsilon-delta condition from metric spaces without relying directly on inverse images of open sets. Another characterization employs sequences, particularly in spaces with sufficient countability. A function f: X \to Y is sequentially continuous at x \in X if, whenever a sequence \{x_n\} in X converges to x, the sequence \{f(x_n)\} in Y converges to f(x). In any topological space, continuity implies sequential continuity, but the converse holds if and only if X is first-countable, meaning every point has a countable local basis.[55] For example, in metric spaces, which are first-countable, these notions coincide, allowing sequential limits to fully capture topological continuity. To extend this to arbitrary topological spaces, nets provide a generalization of sequences. A net in X is a function from a directed set to X, and it converges to x \in X if every neighborhood of x eventually contains all net values beyond some index. A function f: X \to Y is continuous at x if and only if, for every net \{x_\alpha\} in X converging to x, the net \{f(x_\alpha)\} in Y converges to f(x).[54] This characterization works universally, as nets detect the topology in non-first-countable spaces where sequences may fail, such as the product topology on uncountable products. Filters offer yet another equivalent perspective, generalizing both sequences and nets through ultrafilter-like structures. A filter on X converges to x \in X if every neighborhood of x belongs to the filter. The function f: X \to Y is continuous at x if and only if, for every filter \mathcal{F} on X converging to x, the image filter f(\mathcal{F}) = \{f(A) \mid A \in \mathcal{F}\} converges to f(x) in Y.[56] This approach is particularly useful in spaces where convergence needs to be defined without ordering, and it aligns with the neighborhood filter at x, which always converges to x. Finally, continuity can be expressed using closure operators. The closure \mathrm{cl}_X(A) of a subset A \subseteq X is the smallest closed set containing A. The function f: X \to Y is continuous if and only if, for every subset A \subseteq X, f(\mathrm{cl}_X(A)) \subseteq \mathrm{cl}_Y(f(A)).[57] This condition reflects how continuous maps preserve limits of sets, ensuring that images of adherent points remain adherent, and it holds globally without reference to points or open sets.Homeomorphisms
A homeomorphism between two topological spaces X and Y is a bijective continuous map f: X \to Y whose inverse f^{-1}: Y \to X is also continuous.[58] This condition ensures that f preserves the topological structure of open sets, meaning f maps open sets in X to open sets in Y, and equivalently for f^{-1}.[58] Two spaces are said to be homeomorphic, denoted X \cong Y, if such a map exists, establishing topological equivalence where the spaces are indistinguishable by any topological property.[59] Examples of homeomorphisms abound in familiar spaces. On smooth manifolds, a diffeomorphism—a bijective smooth map with a smooth inverse—is necessarily a homeomorphism, as smoothness implies continuity.[60] A concrete instance is the homeomorphism between the real line \mathbb{R} and the open interval (-\pi/2, \pi/2), given by the tangent function \tan: (-\pi/2, \pi/2) \to \mathbb{R}, which is bijective, continuous, and has a continuous inverse \arctan: \mathbb{R} \to (-\pi/2, \pi/2).[58] This extends to any open interval (a, b) via a scaled and translated version, such as f(x) = \tan\left(\pi \frac{x - (a+b)/2}{b-a}\right), confirming that all bounded open intervals on \mathbb{R} are homeomorphic to the entire line.[58] Homeomorphisms preserve topological invariants—properties invariant under such maps—allowing classification of spaces up to topological equivalence. Key preserved properties include compactness, as the continuous image of a compact set is compact and bijectivity ensures the inverse behaves similarly; connectedness, where the image of a connected space remains connected; and separation axioms like Hausdorffness.[58] In contrast, properties beyond topology, such as differentiability or metric distances, are not preserved; for instance, the tangent map above is not differentiable at the endpoints in an extended sense, despite being a homeomorphism.[60] Homeomorphisms facilitate construction via gluing: if compatible open covers of spaces yield homeomorphic local pieces, the global glued space inherits the homeomorphism. Specifically, the pasting lemma ensures that continuous bijections on overlapping open sets combine to a homeomorphism on the union, as seen in gluing charts for manifolds like the Grassmannian, where local Euclidean pieces are homeomorphic under compatible transitions.[61] Thus, if two spaces are homeomorphic and glued along homeomorphic subsets via compatible maps, the resulting spaces are homeomorphic.[61]Core topological properties
Compactness
In general topology, a topological space X is defined to be compact if every open cover of X has a finite subcover.[62] This property captures a form of "finiteness" in infinite spaces, generalizing the behavior of finite sets where any cover trivially admits a finite subcover. Compactness is a topological invariant, preserved under homeomorphisms, and serves as a foundational concept for many theorems in analysis and geometry. In metric spaces, compactness implies that the space is both closed and bounded, though the converse does not hold in general.[63] Specifically, the Heine-Borel theorem establishes that for subsets of \mathbb{R}^n with the standard topology, a set is compact if and only if it is closed and bounded. However, in arbitrary Hausdorff spaces or non-metric topologies, closed and bounded sets need not be compact; counterexamples include certain infinite-dimensional normed spaces where bounded closed balls fail to be compact. Compact sets in Hausdorff spaces are always closed, as the complement of a compact set is open.[64] A key property of compact spaces is that continuous images of compact spaces are compact. If f: X \to Y is a continuous function and X is compact, then f(X) is compact in Y. This preservation under continuous maps underscores compactness's role in ensuring extremal values, such as the extreme value theorem for continuous functions on compact subsets of \mathbb{R}^n. Sequential compactness provides an equivalent characterization in metric spaces: a metric space is compact if and only if every sequence in the space has a convergent subsequence. In general topological spaces, sequential compactness (every sequence has a convergent subsequence) implies compactness but not conversely, as there exist compact spaces without this sequential property, such as certain uncountable products. Local compactness is a related but weaker notion: a topological space is locally compact if every point has a neighborhood basis consisting of compact sets. Every compact space is locally compact, but the converse fails; for example, the real line \mathbb{R} is locally compact but not compact overall.[65] Locally compact Hausdorff spaces admit useful one-point compactifications, extending to non-compact cases like \mathbb{R}^n. Tychonoff's theorem states that the product of any collection of compact topological spaces, equipped with the product topology, is compact.[66] This result, relying on the axiom of choice, enables the compactness of infinite products like the Hilbert cube [0,1]^\mathbb{N}, with applications in functional analysis.Connectedness
A topological space X is said to be connected if it cannot be expressed as the union of two nonempty disjoint open subsets whose union is X.[67] Equivalently, X is connected if the only clopen subsets (subsets that are both open and closed) of X are the empty set and X itself.[68] This property captures the intuitive notion of a space being "in one piece," preventing it from being split into separated parts by the topology.[68] The connected components of a topological space X are the maximal connected subsets of X.[69] For each point x \in X, the connected component containing x, denoted C(x), is the largest connected subset of X that includes x, and it can be constructed as the intersection of all connected subsets containing x.[67] These connected components form a partition of X, meaning every point belongs to exactly one component, and distinct components are disjoint.[69] A space X is locally connected at a point x if every neighborhood of x contains a connected neighborhood of x, and X is locally connected if it is locally connected at every point; in such spaces, the connected components are open subsets.[67] A stronger notion is path-connectedness: a topological space X is path-connected if, for any two points x, y \in X, there exists a continuous function \gamma: [0,1] \to X such that \gamma(0) = x and \gamma(1) = y.[70] Path-connectedness implies connectedness, since if X = U \cup V were a separation into nonempty disjoint open sets, a path from a point in U to a point in V would have to jump discontinuously between them, contradicting continuity.[70] However, the converse does not hold; the topologist's sine curve provides a classic counterexample. Define S = \{(x, \sin(1/x)) \mid 0 < x \leq 1\} \cup \{(0, y) \mid -1 \leq y \leq 1\} with the subspace topology from \mathbb{R}^2. This space S is connected because its only connected subsets are either contained in the vertical segment at x=0 or include points from the sine curve, preventing a separation, but it is not path-connected since no continuous path can connect a point on the vertical segment (away from the origin) to a point on the oscillating curve without violating the uniform continuity bound on paths.[70] Examples of disconnected spaces abound. The set of rational numbers \mathbb{Q}, endowed with the subspace topology from \mathbb{R}, is disconnected; for instance, it can be separated into \mathbb{Q} \cap (-\infty, \sqrt{2}) and \mathbb{Q} \cap (\sqrt{2}, \infty), both nonempty and open in \mathbb{Q}.[43] In fact, \mathbb{Q} is totally disconnected, meaning its only connected subsets are singletons, as any two distinct rationals can be separated by open intervals avoiding irrationals between them.[43] Discrete spaces, where every subset is open, are also totally disconnected, with each singleton as a component.[69] Connectedness is preserved under continuous maps: the continuous image of a connected space is connected.[67] For example, the real line \mathbb{R} with the standard topology is connected, as any separation would contradict the intermediate value theorem for continuous functions on intervals.[67] Variants like the topologist's sine curve illustrate spaces that are connected but lack path-connectedness, highlighting the subtlety between these notions.[70]Separation and countability
Separation axioms
Separation axioms form a hierarchy of conditions on topological spaces that quantify the extent to which distinct points or closed sets can be distinguished using open neighborhoods. These properties, developed primarily in the early 20th century, are essential for ensuring well-behaved convergence, continuity, and embedding into metric spaces.[71] The axioms range from weak separation, where points are minimally distinguishable, to stronger ones that allow separation of closed sets, facilitating applications in analysis and geometry.[71] The T0 axiom, also called the Kolmogorov axiom after Andrey Kolmogorov's 1937 work on topological invariants, requires that for any two distinct points x, y in a space X, there exists an open set containing one but not the other.[72] This ensures points are topologically distinguishable in at least one direction, though not symmetrically. Examples include the Zariski topology on algebraic varieties, where non-T0 spaces can arise but are often quotiented to T0 versions for uniqueness.[71] A T1 space, named after Maurice Fréchet's 1906 thesis on functional calculus, strengthens T0 by requiring that singletons are closed sets, or equivalently, that every pair of distinct points has open neighborhoods excluding the other.[11] In T1 spaces, finite sets are closed, and limits of sequences, if they exist, are unique. The cofinite topology on an infinite set exemplifies a T1 space: open sets are those with finite complements, making singletons closed but failing stronger separation.[73] The T2 axiom, or Hausdorff axiom from Felix Hausdorff's 1914 foundational text on set theory, demands that distinct points possess disjoint open neighborhoods.[74] This symmetric separation implies unique sequential limits and is standard in most practical topologies, such as Euclidean spaces. Non-Hausdorff examples like the cofinite topology highlight pathologies, such as non-unique limits, underscoring T2's importance for analysis.[71] T3 spaces, often defined as T1 plus regular (where points and disjoint closed sets have disjoint open neighborhoods), allow separation of points from closed sets not containing them.[73] Regularity ensures closed sets behave well under continuous functions. Some conventions merge T3 with T0 instead of T1, but the T1 version is common for Hausdorff-like progression.[71] The strongest common axiom, T4 or normal, combines T1 with the ability to separate any two disjoint closed sets by disjoint open sets.[73] Normality enables the existence of continuous functions separating closed sets, as in Urysohn's lemma. A key implication is Urysohn's metrization theorem: a second-countable T3 space is metrizable, embedding it into a metric space while preserving topology.[75] This bridges abstract topology to metric analysis, with examples like manifolds benefiting from such structure.[71] Weaker axioms like T0 appear in algebraic geometry, while stronger ones like T4 dominate in differential topology; enhancements with countability axioms yield even finer classifications.[72]Countability axioms
In general topology, countability axioms impose restrictions on the "size" of the topology by requiring certain countable structures, such as bases or dense subsets, which facilitate the use of sequences and countable covers in proofs of continuity and compactness-like properties. These axioms are particularly useful in distinguishing metrizable spaces and ensuring that abstract topological concepts behave similarly to those in familiar Euclidean spaces.[76] A topological space X is first-countable if, for each point x \in X, there exists a countable local basis \{B_n(x)\}_{n \in \mathbb{N}} consisting of neighborhoods of x such that every neighborhood of x contains some B_n(x). In such spaces, sequences suffice to characterize continuity and limits: a function f: X \to Y is continuous at x if and only if, for every sequence \{x_n\} in X converging to x, the sequence \{f(x_n)\} converges to f(x) in Y. First-countability holds in all metrizable spaces, as the balls of rational radii around each point form a countable local basis.[76] A topological space X is second-countable if it admits a countable basis \mathcal{B} = \{U_n\}_{n \in \mathbb{N}} for its topology, meaning every open set in X is a union of elements from \mathcal{B}. Every second-countable space is first-countable, as the collection of basis elements containing a fixed point x provides a countable local basis at x. The real line \mathbb{R} with the standard topology is second-countable, with the open intervals having rational endpoints forming a countable basis. Subspaces and countable products of second-countable spaces remain second-countable.[76] A topological space X is separable if it contains a countable dense subset D \subseteq X, meaning the closure of D is all of X. Second-countable spaces are necessarily separable: given a countable basis \mathcal{B}, select one point from each non-empty basis element to form a countable dense set (using the axiom of countable choice). However, separability does not imply second-countability in general; for example, the lower limit topology on \mathbb{R} (also known as the Sorgenfrey line) is separable but not second-countable. An uncountable discrete space, where every subset is open, cannot be separable, as any dense subset would need to intersect every singleton and thus be uncountable.[76] A topological space X is Lindelöf if every open cover of X admits a countable subcover. Second-countable spaces are Lindelöf, since any open cover can be refined to the countable basis, and the basis elements used in the unions suffice as a countable subcover. The product of two Sorgenfrey lines is not Lindelöf, despite each factor being Lindelöf, illustrating that the property is not preserved under arbitrary products. First-countable spaces need not be Lindelöf, but in metric spaces, separability, Lindelöf, and second-countability are equivalent.[76] The implications among these axioms form a hierarchy: second-countable implies first-countable (and thus sequential, where the topology is determined by sequential convergence), which in turn implies properties like the sequential characterization of closed sets, but none of these fully imply separability or Lindelöf without additional assumptions. These countability conditions interact with separation axioms by ensuring that points can be distinguished using countable structures, though they primarily address covering and density rather than point separation.[76] A topological space X satisfies the countable chain condition (ccc) if every collection of pairwise disjoint non-empty open sets in X is at most countable. Separable spaces satisfy the ccc, as a countable dense set intersects every non-empty open set, limiting the size of disjoint families. However, the ccc does not imply separability; for instance, the Cantor cube \{0,1\}^\kappa for uncountable \kappa with the product topology has the ccc but is not separable. The ccc is a cardinal restriction on the topology, often used in studying compactness and covering properties in non-metrizable spaces.[77]Metric and uniform structures
Metric spaces
A metric space is a pair (X, d), where X is a set and d: X \times X \to [0, \infty) is a function, called a metric, satisfying the following axioms for all x, y, z \in X:- d(x, y) = 0 if and only if x = y (identity of indiscernibles),
- d(x, y) = d(y, x) (symmetry),
- d(x, z) \leq d(x, y) + d(y, z) (triangle inequality).[78]