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A set U in a topological space (X, τ) is open if U ∈ τ. Open sets are the building blocks of a topology and generalize open intervals from ℝ. They are defined by closure under arbitrary unions and finite intersections rather than by distance.
When you study real analysis, an open set in ℝ is typically defined as a set where every point has some open interval around it that stays inside the set — the set (0, 1) is open because for any x ∈ (0, 1), you can find ε > 0 so that (x−ε, x+ε) ⊆ (0, 1). This definition relies entirely on distance. Topology asks: what if we strip away distance but keep the essential structure that makes "open" useful? The answer is to simply declare which sets count as open.
A topology on a set X is a collection τ of subsets of X satisfying three axioms: (1) ∅ and X are in τ; (2) any union of sets in τ is in τ; (3) any finite intersection of sets in τ is in τ. A set is open if and only if it belongs to τ — that is the entire definition. Notice there is no mention of distance, neighborhoods, or real numbers. The axioms capture the algebraic behavior of open sets in ℝ while leaving the concept free to apply to any set at all.
The two extreme topologies illustrate the range of possibilities. The indiscrete topology τ = {∅, X} is the coarsest: only the empty set and the whole space are open. The discrete topology τ = P(X) is the finest: every subset is open. Most useful topologies live between these extremes. The standard topology on ℝ — where open sets are arbitrary unions of open intervals — is one such topology, and it can be recovered from the axiomatic definition by verifying the three properties hold.
Why only *finite* intersections? If you allow infinite intersections, open sets are no longer stable under intersection: in ℝ, the sets (-1/n, 1/n) are all open, but their intersection is {0}, which is not open. The axiom is designed to exclude this case. Arbitrary unions are allowed because union only makes sets bigger, which does not cause similar problems. These asymmetries — arbitrary unions, finite intersections — reappear constantly in topology and are worth memorizing.
Open sets are the raw material from which all other topological concepts are built. Closed sets are defined as complements of open sets. Continuity of a function f: X → Y means the preimage of every open set in Y is open in X. Compactness, connectedness, and convergence are all ultimately defined in terms of open sets. This is why getting comfortable with what "open" means at the axiomatic level — membership in τ, not proximity — is so foundational.