A topological space is a set X together with a collection τ of subsets of X — called the open sets — satisfying three axioms: (1) the empty set and X itself are in τ, (2) any union of sets in τ is in τ, and (3) any finite intersection of sets in τ is in τ.
The collection τ is called a topology on X. The same set can carry different topologies: the discrete topology (every subset is open), the indiscrete topology (only ∅ and X are open), and everything in between. The choice of topology determines which functions are continuous, which sequences converge, and what “nearness” means — all without reference to distance.
A topological space viewed as a category of open sets (with inclusions as morphisms) is the starting point for sheaf theory: a presheaf assigns data to each open set, and a sheaf is a presheaf whose local data glues uniquely. A topological space can also be described as a frame — a complete lattice where finite meets distribute over arbitrary joins — and this algebraic description generalizes to locales, which do not require an underlying set of points.
The open sets of a topological space form a complete Heyting algebra: meet is intersection, join is union, and the Heyting implication U → V is the interior of (V ∪ Uᶜ). This is the prototypical example of the algebraic structure underlying the semiotic universe — the modality j generalizes the topological closure operator.