Glossary of Topology - W

W

Weak topology

The weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the coarsest topology on the set which makes all the functions continuous.

Weaker topology

See Coarser topology. Beware, some authors, especially analysts, use the term stronger topology.

Weakly countably compact

A space is weakly countably compact (or limit point compact) if every infinite subset has a limit point.

Weakly hereditary

A property of spaces is said to be weakly hereditary if whenever a space has that property, then so does every closed subspace of it. For example, compactness and the Lindelöf property are both weakly hereditary properties, although neither is hereditary.

Weight

The weight of a space X is the smallest cardinal number κ such that X has a base of cardinal κ. (Note that such a cardinal number exists, because the entire topology forms a base, and because the class of cardinal numbers is well-ordered.)

Well-connected

See Ultra-connected. (Some authors use this term strictly for ultra-connected compact spaces.)