Glossary of topology

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology. All spaces in this glossary are assumed to be topological spaces unless stated otherwise. Absolutely closed See H-closed Accessible See . Accumulation point See limit point. Alexandrov topology The topology of a space X is an Alexandrov topology (or is finitely generated) if arbitrary intersections of open sets in X are open, or equivalently, if arbitrary unions of closed sets are closed, or, again equivalently, if the open sets are the upper sets of a poset. Almost discrete A space is almost discrete if every open set is closed (hence clopen). The almost discrete spaces are precisely the finitely generated zero-dimensional spaces. α-closed, α-open A subset A of a topological space X is α-open if , and the complement of such a set is α-closed. Approach space An approach space is a generalization of metric space based on point-to-set distances, instead of point-to-point. Baire space This has two distinct common meanings: A space is a Baire space if the intersection of any countable collection of dense open sets is dense; see Baire space. Baire space is the set of all functions from the natural numbers to the natural numbers, with the topology of pointwise convergence; see Baire space (set theory). Base A collection B of open sets is a base (or basis) for a topology if every open set in is a union of sets in . The topology is the smallest topology on containing and is said to be generated by . Basis See Base. β-open See Semi-preopen. b-open, b-closed A subset of a topological space is b-open if . The complement of a b-open set is b-closed. Borel algebra The Borel algebra on a topological space is the smallest -algebra containing all the open sets. It is obtained by taking intersection of all -algebras on containing .