Topologised

Topologised is the past tense of topologise (British English) or topologize (American English). In mathematics, to topologise a set means to endow it with a topology, thereby turning it into a topological space. A topology on a set X is a collection T of subsets of X, called open sets, that satisfies three conditions: the empty set and X are in T; the union of any collection of members of T is also in T; and the intersection of any finite number of members of T is in T. When these conditions hold, the pair (X, T) is a topological space.

Common examples illustrate the idea. The real numbers R with the standard topology uses open sets that can be expressed as unions of open intervals (a, b). The discrete topology on a set X is obtained by taking T to be the power set of X, so every subset is open. The indiscrete (or trivial) topology on X has only ∅ and X open. These examples show how different choices of T encode notions of openness, continuity, and convergence on the same underlying set.

The act of topologising is essential for studying properties that depend on the notion of closeness or continuity, such as continuity of functions, convergence of sequences, and connectedness, up to homeomorphism. In literature, topologised can simply mean “endowed with a topology” or “considered as a topological space.” The term is commonly used in mathematical contexts, with the British spelling topologise and the American spelling topologize both in circulation.