Open set

An open set is a subset of a topological space in which every point has a neighborhood contained in the set. Openness depends on the topology, not just the points.

Topology: A structure specifying which subsets of a set are open and thereby defining continuity and neighborhoods. Open sets are the defining data of a topology.
Topological space: A set together with a collection of subsets designated as open, satisfying the topology axioms. Its axioms govern unions and finite intersections of open sets.
Convergence in a topological space: Convergence defined by eventual membership in every neighborhood of a proposed limit. Open neighborhoods provide the tests that determine topological convergence.
Discrete topology: The topology in which every subset of a set is open. It makes openness automatic for every subset.
Neighborhood (topology): A set containing an open set around a specified point. A point belongs to an open set exactly when that set is one of its neighborhoods.
Union (set theory): The set of elements belonging to at least one member of a collection of sets. Any union of open sets remains open, even for infinitely many sets.
Homeomorphism: A bijective continuous map whose inverse is also continuous. Homeomorphisms preserve open sets and capture topological sameness.
Indiscrete topology: The topology containing only the empty set and the whole underlying set. It permits almost no open subsets, unlike the discrete topology.
Interior (topology): The largest open subset contained in a given subset of a topological space. It collects precisely the points whose neighborhoods fit inside the set.
Intersection (set theory): The set of elements belonging to every member of a collection of sets. Finite intersections of open sets remain open, while infinite ones need not.

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