A topological space (X, τ) is a set X together with a collection τ of subsets (open sets) satisfying three axioms: ∅ and X are in τ; arbitrary unions of sets in τ remain in τ; finite intersections of sets in τ remain in τ. This axiomatizes 'openness' from real analysis, providing a framework for continuity and convergence without requiring distance metrics.
A topological space is a pair (X, τ) where X is a set and τ is a collection of subsets of X — called the open sets — satisfying three axioms: (1) the empty set ∅ and the whole set X belong to τ, (2) the union of any collection of sets in τ belongs to τ, and (3) the intersection of any finite collection of sets in τ belongs to τ. These axioms distill the essential properties of "openness" from real analysis and metric spaces into a minimal framework that requires no notion of distance. Everything in topology — continuity, convergence, compactness, connectedness — is built on this foundation.
The axioms are modeled on the behavior of open sets in ℝ. In the standard topology on ℝ, unions of open intervals are open, and finite intersections of open intervals are open. The topology axioms abstract exactly these closure properties. The asymmetry between arbitrary unions and finite intersections is not accidental: in ℝ, the infinite intersection ∩ₙ(−1/n, 1/n) = {0} is not open, so requiring closure under arbitrary intersections would exclude the standard topology. The finite-intersection restriction is precisely what allows the standard topology on ℝ to qualify while preventing a collapse to the discrete topology.
Any set X admits multiple topologies. The discrete topology τ = P(X) declares every subset open; the indiscrete topology τ = {∅, X} declares only ∅ and X open. These are the finest and coarsest possible topologies on X. Between them, many intermediate topologies exist. On ℝ, beyond the standard topology, one can define the cofinite topology (open sets are those with finite complement, plus ∅), the lower-limit topology (basis of half-open intervals [a, b)), and others. Each choice of τ yields a different topological space with different notions of convergence, continuity, and compactness — even though the underlying set X is the same.
The power of the axiomatic approach is its generality. Once the three axioms are in place, you can define continuity (a function f : X → Y is continuous if f⁻¹(U) is open in X whenever U is open in Y), convergence (a sequence converges to x if it is eventually in every open set containing x), compactness (every open cover has a finite subcover), and connectedness (no nontrivial partition into disjoint open sets) — all without mentioning distance. Metric spaces become a special case: every metric induces a topology via open balls, and all metric-space theorems about continuity and convergence are instances of the general topological definitions. The passage from metric spaces to topological spaces is the passage from quantitative distance to the qualitative structure of "which sets are open," and the three axioms are all that structure requires.