finally compact

Definition

A topological space is called finally compact if any open cover of this space contains a countable subcover.

Property

• Every Lindeloef space is strongly paracompact.
• Every separable metacompact space is finally compact.
• For separable T_3-spaces, metacompactness, paracompactness, strong paracompactness and final compactness are equivalent.
• A connected T_3-space is strongly paracompact if and only if it is finally compact.

Remark

• This space is usually called Lindeloef but sometimes finally compact regular space is called a Lindeloef space. The term "Lindeloef" above stands for finally compact regular spaces.

Reference

A.V. Arhangel'skii (ed), General topology III: paracompactness, function spaces, descriptive theory, Springer (1989)