Topology

Specifying every single open set of a topology directly is often unwieldy, especially for infinite spaces. A basis is a smaller, more manageable collection of subsets — basis elements — from which the entire topology can be reconstructed as all possible unions of basis elements. For example, the open intervals of ℝ form a basis for the standard topology: instead of listing every open subset of ℝ (an intractable task), you only need to describe the intervals, and every open set is automatically a union of some collection of them. A basis is characterized by two simple conditions: the basis elements must cover the whole space, and the intersection of any two basis elements must itself contain a basis element around every one of its points.

In a general topological space, compactness is defined via open covers, but in a metric space this abstract condition turns out to be equivalent to two more concrete, hands-on properties: sequential compactness (every sequence has a subsequence converging to a point of the space) and the combination of completeness and total boundedness. This three-way equivalence — compactness ⟺ sequential compactness ⟺ complete + totally bounded — is one of the central theorems of metric space theory, and it specializes further in Euclidean space ℝⁿ to the Heine–Borel theorem: compact ⟺ closed and bounded. The equivalence lets analysts freely switch between the cover-based definition (good for proving abstract theorems) and the sequence-based definition (good for concrete computations) whenever a metric is present.

A topological space X is connected if it cannot be split into two disjoint, nonempty open sets whose union is all of X — such a splitting is called a separation. Intuitively, a connected space is 'all one piece': there is no way to draw a clean open-set boundary dividing it into two genuinely separate parts. A stronger and often more intuitive notion, path-connectedness, requires that any two points can be joined by a continuous path inside the space; every path-connected space is connected, though the converse can fail. Connectedness is the topological property underlying the Intermediate Value Theorem: a continuous real-valued function on a connected domain cannot skip over values.

Continuity is usually first learned via the ε-δ definition on ℝ: f is continuous at x if points close to x map to points close to f(x). Topology strips this down to its essence, defining continuity purely in terms of open sets, with no distance required: a function f: X → Y between topological spaces is continuous if the preimage of every open set in Y is open in X. This single, elegant condition recovers the ε-δ definition exactly when X and Y are metric spaces (in particular ℝ), and extends continuity to arbitrary topological spaces where no notion of distance exists at all.

A topological space is Hausdorff (or T2) if any two distinct points can be separated by disjoint open sets — a neighborhood around each point that doesn't overlap with the other's. This mild-sounding separation axiom, named for Felix Hausdorff who used it as part of his original definition of a topological space, rules out pathological behavior that would be strange in any space thought of as a generalization of ordinary geometry: in a Hausdorff space, limits of sequences (and more general nets) are unique, and compact subsets are automatically closed. Nearly every space encountered in analysis and geometry — metric spaces, manifolds, ℝⁿ — is Hausdorff; non-Hausdorff spaces are the exception, used mainly as deliberately pathological examples or in specialized settings like algebraic geometry's Zariski topology.

A metric space is a set X equipped with a distance function d: X × X → ℝ that assigns a nonnegative real number to every pair of points, measuring how 'far apart' they are. This distance function, called a metric, must satisfy three natural axioms: identical points have distance zero (and only identical points do), distance is symmetric, and distances satisfy a triangle inequality. Metric spaces give the most familiar route into topology, because every metric induces a topology — a collection of open sets built from open balls — recovering the usual ε-neighborhood notion of closeness. Many different metrics can induce the very same topology, which is exactly the sense in which topology is a strictly more general framework than metric geometry.

Given two topological spaces X and Y, the product topology equips the Cartesian product X × Y with a natural topology built from a basis of 'rectangles' U × V, where U is open in X and V is open in Y. This construction generalizes the familiar plane ℝ² = ℝ × ℝ, whose standard topology (generated by open rectangles) coincides with the product topology built from ℝ's own standard topology. For products of infinitely many spaces, there are two competing generalizations — the box topology and the product topology — and they differ: the product topology, the one that matters throughout most of mathematics (and the one for which Tychonoff's theorem holds), only restricts finitely many coordinates at a time, leaving all remaining coordinates completely unconstrained.

The quotient topology is the standard way to build new topological spaces by 'gluing together' points of an existing space according to some equivalence relation. Given a space X and an equivalence relation ~ on X, the quotient set X/~ collects points into equivalence classes, and the quotient map q: X → X/~ sends each point to its class. The quotient topology declares a subset U of X/~ to be open exactly when its preimage q⁻¹(U) is open back in X — this is the unique topology on X/~ making q continuous in the strongest possible sense compatible with the gluing. This construction produces many familiar spaces from simple building blocks: identifying the two endpoints of an interval produces a circle, and identifying antipodal points of a sphere produces the projective plane.

A topological space is, roughly speaking, a space in which 'closeness' is defined but cannot necessarily be measured by a numeric distance. Formally, it is a set X together with a collection τ of subsets of X, called the open sets, satisfying three axioms: the empty set and X itself are open, arbitrary unions of open sets are open, and finite intersections of open sets are open. This single structure — a topology — is enough to define continuity, convergence, connectedness, and compactness without ever mentioning a distance function, making topology the natural generalization of the geometry of the real line and Euclidean space to arbitrarily abstract sets.

In a topological space (X, τ), the open sets are by definition exactly the members of τ, and a closed set is defined as a set whose complement in X is open. Open and closed sets are generalizations of the open interval (a,b) and closed interval [a,b] on the real line — but unlike common English usage of 'open' and 'closed' as opposites, a set can be both open and closed (clopen) or neither. Open sets formalize 'having wiggle room around every point'; closed sets formalize 'containing all of their own limit points.' Together they are the basic vocabulary from which continuity, closure, boundary, and compactness are all built.

Compactness is a property of a topological space that makes it behave in many ways like a finite set, even when it has infinitely many points. The general (open-cover) definition says a space is compact if every collection of open sets that covers the space has a finite sub-collection that still covers it. For subsets of Euclidean space, this is equivalent to sequential compactness — every infinite sequence in the set has a subsequence converging to a point of the set — and, by the Heine–Borel theorem, to simply being closed and bounded. Just as every real-valued function on a finite set is bounded and attains its maximum and minimum, every continuous real-valued function on a compact space has these same properties (the extreme value theorem).

Given a subset A of a topological space X, three companion sets describe how A sits inside X: the interior of A (the largest open set contained in A), the closure of A (the smallest closed set containing A), and the boundary of A (the points that are 'on the edge' — in the closure of both A and its complement). These three constructions recover, in any topological space, exactly the everyday geometric intuition of interior, closure, and edge for shapes in the plane, while also making sense for wild, non-metric spaces. Every point of X falls into exactly one of: the interior of A, the interior of the complement of A, or the boundary of A — a three-way partition of the whole space relative to A.

Given a topological space (X, τ) and a subset Y ⊆ X, the subspace topology on Y is the natural way to make Y itself into a topological space, using X's topology as a guide: a subset of Y is declared open exactly when it is the intersection of Y with some open set of X. This is how every subset of ℝⁿ inherits a topology — for instance, a closed interval [a,b], a circle, or a Cantor set are each viewed as topological spaces via the subspace topology from the ambient Euclidean space. A key subtlety is that 'open in Y' does not mean 'open in X': a set can be open in the subspace topology on Y while not being open in X itself, precisely when Y is not open in X.

A topological space X is path-connected if any two points x, y ∈ X can be joined by a path — a continuous function γ: [0,1] → X with γ(0) = x and γ(1) = y. This is a stronger, more hands-on notion than plain connectedness: instead of merely forbidding a clean open/open split of the space, path-connectedness demands you can actually walk continuously from any point to any other while staying inside X. Every path-connected space is connected, but the converse fails — the topologist's sine curve is the classic counterexample: connected, yet with no path joining a point on its oscillating part to a point on its limiting vertical segment. Path-connectedness is also the foundation for the fundamental group, since loops (paths starting and ending at the same point) are exactly what π₁ organizes into a group.

The separation axioms are a hierarchy of conditions (T0, T1, T2, T3, T4, ...) that a topological space may or may not satisfy, each demanding progressively stronger ways of 'telling points or sets apart' using open sets. T0 (Kolmogorov) merely asks that for any two distinct points, some open set contains one but not the other; T1 strengthens this to say singletons are closed; T2 (Hausdorff) demands disjoint open neighborhoods for distinct points; T3 (regular, plus T1) separates points from closed sets not containing them; and T4 (normal, plus T1) separates disjoint closed sets from each other. Each axiom in the hierarchy strictly implies the ones before it (T4+T1 ⟹ T3+T1 ⟹ T2 ⟹ T1 ⟹ T0), and standard counterexamples show none of the reverse implications hold — the hierarchy genuinely stratifies how 'nicely separated' a space's points and closed sets are.

A compact Hausdorff space is a topological space that is simultaneously compact (every open cover has a finite subcover) and Hausdorff (distinct points have disjoint open neighborhoods). This combination is one of the most useful in all of topology: compactness alone provides finiteness-like control, and Hausdorffness alone provides separation, but together they force a rigidity that neither has on its own. In a compact Hausdorff space, compact subsets are automatically closed, the space is automatically normal (T4), and — most famously — any continuous bijection from a compact space onto a Hausdorff space is automatically a homeomorphism. This last fact means that on compact Hausdorff spaces, the topology is 'as fine as it can be while staying compact, and as coarse as it can be while staying Hausdorff': you cannot add open sets without destroying compactness, and you cannot remove any without destroying the Hausdorff property.

The fundamental group π₁(X, x₀) is one of the first and most important invariants in algebraic topology: it converts a topological question ('are these spaces really the same shape?') into an algebraic one ('are these groups isomorphic?'). It is built from loops — continuous paths that start and end at a fixed basepoint x₀ — where two loops are considered equivalent if one can be continuously deformed into the other without leaving the space (a homotopy). The set of these equivalence classes forms a group under concatenation of loops, called the fundamental group. Crucially, homeomorphic spaces have isomorphic fundamental groups, which makes π₁ a powerful tool for proving spaces are topologically different, even when they look superficially similar.

A topological invariant is any property or algebraic object attached to a topological space that is preserved by homeomorphism — that is, if X and Y are homeomorphic, they must have the same value of the invariant. Invariants are the primary tool for proving that two spaces are NOT homeomorphic: since a homeomorphism must preserve the invariant, spaces with different invariant values simply cannot be topologically the same, no matter how one tries to bend or stretch one into the other. Key examples include connectedness, compactness, the Euler characteristic, and the fundamental group; more refined invariants (homology and cohomology groups) extend this idea to distinguish spaces that cruder invariants cannot tell apart.

A topological group is a set that is simultaneously a group and a topological space, with the two structures made compatible by requiring the group operations — multiplication and inversion — to be continuous. This marriage of algebra and topology is ubiquitous: (ℝ, +) with its usual topology, the circle S¹ = U(1) under multiplication of unit complex numbers, and the general linear group GL(n, ℝ) of invertible matrices under matrix multiplication are all topological groups. The continuity requirement lets one import topological tools (compactness, connectedness, continuity) into group theory and vice versa: a topological group's connected component of the identity is itself a subgroup, compact topological groups carry a canonical translation-invariant (Haar) measure, and Lie groups — topological groups that are also smooth manifolds — are the central objects of study in continuous symmetry, underlying everything from crystallography to gauge theories in physics.

A covering space of a topological space X is a space X̃ together with a continuous surjection p: X̃ → X such that every point of X has an open neighborhood U whose full preimage p⁻¹(U) is a disjoint union of open sets, each mapped homeomorphically onto U by p. Intuitively, X̃ 'unrolls' or 'unwinds' X locally without distortion — near any point, X̃ looks like several exact, non-overlapping copies of a neighborhood in X stacked on top of each other. Covering spaces are the geometric backbone of the fundamental group: there is a deep correspondence (the Galois correspondence for covering spaces) between subgroups of π₁(X) and covering spaces of X, with the universal cover — the covering space with trivial fundamental group — corresponding to the trivial subgroup and covering every other connected covering space of X.

A homotopy is a continuous deformation of one continuous map into another, and two spaces are homotopy equivalent if each can be continuously deformed into (a space that looks like) the other, even if no actual homeomorphism between them exists. This is a coarser notion of 'sameness' than homeomorphism: a solid disk is not homeomorphic to a single point (they have different cardinalities!), yet the disk is homotopy equivalent to a point, because the disk can be continuously shrunk down to its center. Homotopy equivalence is the natural notion of equivalence for algebraic topology, because all the invariants built from continuous deformation — the fundamental group, homology, cohomology — only see a space up to homotopy equivalence, not up to homeomorphism. Two homotopy equivalent spaces are said to have the same homotopy type.

Characteristic classes are cohomology classes naturally associated to vector bundles that measure their twisting or non-triviality. The three main families are: Stiefel-Whitney classes w_i in H^i(B; Z/2) for real bundles, Chern classes c_i in H^{2i}(B; Z) for complex bundles, and Pontryagin classes p_i in H^{4i}(B; Z) for real bundles viewed via their complexification. They are the primary algebraic invariants distinguishing vector bundles and are central to the Atiyah-Singer index theorem.

Cohomology is the 'dual' theory to homology: instead of mapping simplices into a space, we assign algebraic values to them. The resulting cochain complex yields cohomology groups H^n(X; R) that carry additional structure — most notably a ring structure via the cup product — absent in homology. Cohomology is essential for characteristic classes, de Rham theory, Poincaré duality, and modern algebraic geometry.

A CW complex (Closure-finite, Weak-topology complex) is a type of topological space built by attaching cells of increasing dimension. Introduced by J.H.C. Whitehead, CW complexes provide a flexible and computable framework for algebraic topology: every manifold is homotopy equivalent to a CW complex, and the cellular chain complex computes homology efficiently. The 'C' stands for closure-finite (each cell meets only finitely many other cells) and 'W' for weak topology.

De Rham cohomology computes the cohomology of a smooth manifold using differential forms rather than singular cochains. The k-th de Rham cohomology group H^k_{dR}(M) consists of closed k-forms modulo exact k-forms. The de Rham theorem establishes a canonical isomorphism between de Rham cohomology and singular cohomology with real coefficients, connecting analysis and topology.

A fiber bundle is a space that locally looks like a product but may be globally twisted. Formally, a fiber bundle (E, B, pi, F) consists of a total space E, base space B, projection pi: E -> B, and fiber F, such that B has an open cover over which E is homeomorphic to a product. Fiber bundles generalize covering spaces (discrete fiber), vector bundles (vector space fiber), and principal bundles (group fiber), and are fundamental to modern geometry and physics.

The higher homotopy groups pi_n(X, x_0) for n >= 2 generalize the fundamental group by considering continuous maps from the n-sphere S^n into X, modulo homotopy. They are abelian for n >= 2 (unlike pi_1). Computing higher homotopy groups is notoriously difficult — even pi_n(S^2) is unknown for large n — but powerful tools like the long exact sequence of a fibration, the Hurewicz theorem, and spectral sequences provide many results.

Singular homology is a functor from topological spaces to graded abelian groups that measures the presence of 'holes' of various dimensions. The n-th homology group H_n(X) detects n-dimensional holes: H_0 counts connected components, H_1 detects loops not bounding disks, H_2 detects enclosed voids, and so on. It is one of the central tools of algebraic topology, providing invariants that distinguish spaces up to homotopy equivalence.

A vector bundle is a fiber bundle whose fiber is a vector space and whose structure group is the general linear group. Vector bundles arise naturally in differential geometry (tangent and cotangent bundles), algebraic topology (the tautological bundle over projective space), and physics (gauge theory). Their study leads to K-theory, characteristic classes, and index theorems.

A topological n-manifold is a topological space that locally 'looks like' ordinary Euclidean space ℝⁿ, even if its global shape is very different. Formally, it is a Hausdorff, second-countable topological space in which every point has an open neighborhood homeomorphic to an open subset of ℝⁿ. This local-Euclidean condition lets manifolds carry local coordinate systems (charts), while the global topology can be as rich as a sphere, torus, or higher-dimensional analogue. Manifolds are the natural setting for differential geometry and physics — a circle is a 1-manifold, a sphere or torus is a 2-manifold, and the spacetime of general relativity is modeled as a 4-dimensional manifold — because locally Euclidean structure is exactly what is needed to do calculus while still allowing the global topology to be exotic.

A homeomorphism is a continuous, bijective function between two topological spaces whose inverse is also continuous — it's the precise notion of 'the same shape' in topology. Two spaces connected by a homeomorphism are called homeomorphic, and topology studies exactly the properties preserved by such maps (properties invariant under stretching, bending, and twisting, but not tearing or gluing).

A simplicial complex is a combinatorial structure built from vertices, edges, triangles, tetrahedra, and their higher-dimensional analogues (simplices), glued together in a consistent way. Simplicial complexes provide a combinatorial model for topological spaces — every compact polyhedron admits a triangulation as a simplicial complex. They are the foundation of combinatorial topology, computational topology (persistent homology), and the discrete model underlying simplicial sets in homotopy theory.

Two topological spaces are homotopy equivalent if each can be continuously deformed into the other. This is a coarser equivalence than homeomorphism: homotopy equivalent spaces share all homotopy-invariant properties (fundamental group, homology, cohomology) but need not be homeomorphic. A deformation retract is a particularly clean way to see homotopy equivalence: a subspace A \subseteq X is a deformation retract if X can be continuously shrunk onto A while fixing A pointwise. For example, any convex subset of \mathbb{R}^n deformation retracts to a point (making it contractible), and the Mobius band deformation retracts to its core circle S^1.

Sequences are sufficient to describe convergence in metric spaces, but in general topological spaces they fall short: a point may be in the closure of a set without any sequence in the set converging to it. Nets (Moore-Smith sequences) and filters generalise sequences to arbitrary topological spaces, restoring all the familiar sequential intuitions: a set is closed iff it is closed under limits of converging nets, a function is continuous iff it preserves net convergence, and a space is compact iff every net has a convergent subnet. Filters provide an equivalent, more algebraic formulation.