*Definition 1 [#t17ca104]
A topological space is said to be quasicompact if every open cover has a finite subcover.

*Definition 2 [#z94f2300]
A topological space X is said to be quasicompact if every cover of X by [[co-zero]] sets has a finite subcover.

*Remark [#ibc7f310]
-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 [#b3c4aa68]
: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.