quasicompact

Definition 1

A topological space is said to be quasicompact if every open cover has a finite subcover.

Definition 2

A topological space X is said to be quasicompact if every cover of X by co-zero? sets has a finite subcover.

Remark

• For bitopological spaces, see quasi compact.

Definition 1

• This property is often called "compactness". But some authors like Bourbaki use "quasicompact", who includes Hausdorffness in the term "compact".
• This term is often used for non-Hausdorff compact spaces to put stress on non-Hausdorffness. For example, for the spectrum of a ring with Zariski topology.

Definition 2

Reference

Definition 1

• N. Bourbaki, General Topology(Elements of Mathematics), Springer, 2nd printing (1998).
• 上野健爾, 代数幾何, 岩波書店(2005).

Definition 2

• J. K. Hohli and D. Singh, Between compactness and quasicompactness, Acta. Math. Hungar. 106 (4) (2005), 317-329.