*Definition [#r2ef12a9]
A topological space is called finally compact if any open cover of this space contains a countable subcover.

*Property [#efeee300]
-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 [#a015bbf3]
-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 [#u0644940]
A.V. Arhangel'skii (ed), ''General topology III: paracompactness, function spaces, descriptive theory'', Springer (1989)