My definition for "countable set" is a set with the cardinal $\aleph_0$ and "at most countable set" is a set $A$ such that $|A|≦\aleph_0$.
Till now, my definition for Lindelöf space is a topological space which admits a countable subcover of a given open cover.
However, i found this this definition is stronger than that in wikipedia.
In wikipedia, Lindelöf is defined as a space which admits a "at most countable subcover" of a given open cover.
Which one is the usual definition?
Your strict definition cannot even be correct: a finite cover does not have a subcover (in the sense of a subset that is still a cover) that is countably infinite. This already suggests that indeed the meaning "countable = finite or countably infinite", is meant.
If one wants to avoid the "problem" (if we can call it that) we could formulate Lindelöf as: for every open cover $\mathcal{U} = \{ U_i : i \in I \}$ of $X$ there is a function $f: \mathbb{N} \rightarrow I$ such that $X = \cup \{U_{f(n)}: n \in \mathbb{N} \}$ (so that $f$ can even pick the same subset every time, e.g.). But this seems a bit overly formal. I'd prefer: for every cover $\mathcal{U}$ of $X$ there is a $\mathcal{U}' \subseteq \mathcal{U}$ that covers $X$ and $\left|\mathcal{U}'\right| \le \aleph_0$. These amount to the same thing.