Countably Paracompact Lindelöf Space is Paracompact

Theorem
Let $T = \struct {S, \tau}$ be a Lindelöf space which is also countably paracompact.

Then $T$ is paracompact.

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


 * If $T = \struct {S, \tau}$ is a countably paracompact space then every countable open cover of $S$ has an open refinement which is locally finite.

It follows trivially that every open cover of $S$ has an open refinement which is locally finite.

Hence the result by definition of paracompact.