topologized

Topologized is the past participle form of the verb topologize. In mathematics, to topologize a set X means to endow X with a topology, forming a topological space (X, τ). A topology τ is a collection of subsets of X called open sets, satisfying that the empty set and X are in τ, and that τ is closed under arbitrary unions and finite intersections. When a topology is specified, the set X is said to be topologized, or to carry a topology. The term distinguishes a bare set from one that has been equipped with a topological structure. The choice of topology on a given set is not unique; many different topologies may be placed on X, depending on the properties of interest, such as continuity of maps, convergence, or compactness.

Common examples include the discrete topology, where every subset is open, and the trivial (indiscrete) topology, where only ∅ and X are open. In more advanced contexts, one speaks of initial topologies, final topologies, product topologies, or subspace topologies as methods to topologize sets and maps between them in order to control continuity and convergence. In usage, topologized emphasizes that the object is a set equipped with a topology rather than a bare set. See also topology, topological space, topologization, discrete topology, and trivial topology.