topologytype

Topology type is a term used in mathematics to describe the classification of a topological space by the arrangement of its open sets and by the properties that the topology induces. Given a set X, a topology on X is a collection of subsets of X that includes the empty set and X and is closed under arbitrary unions and finite intersections. The particular collection chosen is called the topology, and the term topology type refers to how coarse or fine this collection is and to the standard properties it satisfies. A coarser topology has fewer open sets; a finer topology has more.

Common examples include the discrete topology, where every subset of X is open; the indiscrete (trivial) topology, where only the empty set and X are open; and the cofinite topology, in which open sets are those with finite complement (on infinite X). The standard topology on R, generated by open intervals, is the default topology used in analysis; the lower limit or Sorgenfrey topology on R is generated by half-open intervals [a,b). On a two-point set one can consider the Sierpinski topology, a simple non-discrete example.

Topological spaces are further classified by separation properties (T0, T1, Hausdorff T2, regular, normal), compactness, connectedness, and metrizability (whether a topology arises from a metric). These properties form what is often meant by a topology type in practice. When constructing spaces from others, subspace and product topologies yield new topology types, and quotient topologies can alter separation and compactness properties.

In short, topology type captures the qualitative shape of a space's open sets and the resulting geometric and continuity phenomena.