In the paper

http://topo.math.auburn.edu/tp/reprints/v05/tp05011.pdf

the author claimes (Theorem 2, without proof) that for a completely regular Hausdorff space $X$ the following are equivalent:

(1) The space of continuous functions $C(X)$ with the compact open topology is countably tight.

(2) Every open cover for compact subsets of $X$ has a countable subcover for compact sets.

I try to understand condition (2).

Here an "open cover for compact sets" is an open cover $\mathcal U$ such that every compact subset of $X$ is contained in some member of $\mathcal U$.

Obviously, the second condition looks a lot like the Lindelöf property:

(3) The space $X$ is Lindelöf, i.e. every open cover has a countable subcover.

I think that condition (3) looks stronger than condition (2), because every open cover for compact sets is an open cover and therefore has a countable subcover.

However in the paper it is stated (Proposition 5, also without proof) that (2) implies (3).

(My question is now: Can anybody explain to me what is going on here?)

EDIT: I agree, my question as I wrote it down is not a very well-posed question ;)

Another try: My question consists of two parts:

Is there a space $X$ which satisfies (3), i.e.$X$ is Lindelöf, but does not satisfy (2) ? (this would imply that $C(X)$ is not countably tight with the compact open topology)

Why does (2) imply (3) ? (I know the reference claims that but since there seems to be no proof in the paper, I have problems to believe it)

Thank you in advance! Tom