Countably Compact Lindelöf Space is Compact

Theorem
Let $T = \left({X, \tau}\right)$ be a Lindelöf space which is also countably compact.

Then $T$ is compact.

Proof
By the definitions:
 * If $T = \left({X, \tau}\right)$ is a Lindelöf space then every open cover of $X$ has a countable subcover.


 * If $T = \left({X, \tau}\right)$ is a countably compact space then every countable open cover of $X$ has a finite subcover.

It follows trivially that every open cover of $X$ has a finite subcover.

Hence the result by definition of compact.