An open set in a topological space (X, τ) is any member of the topology τ. Open sets are the primitive notion: they are not defined by a formula but declared by the choice of topology. The axioms require only that τ is closed under arbitrary unions and finite intersections, and contains ∅ and X.
Open sets encode the idea of “observable property” or “verifiable condition.” If a point belongs to an open set, then so do all points sufficiently near it — openness means there is room around every member. In a metric space, open sets are exactly those that contain an open ball around each point, but the topological definition is more general and does not require a metric.
The open sets of a topological space form a lattice: join is union and meet is intersection. Because arbitrary unions are allowed but only finite intersections, this lattice is a frame — and a frame is a complete Heyting algebra. The Heyting implication of open sets U → V is the interior of (V ∪ Uᶜ), the largest open set W such that U ∩ W ⊆ V. This makes every topological space a model of intuitionistic logic, where the modality j (interior of closure) is a Lawvere-Tierney topology.