By "weakly separable" I mean the notion for uniform spaces used by David Wigner and Lawrence Brown: a uniform space is weakly separable if any uniform cover has a countable subcover. For a topological group $G$ this means for any open set $U$ in $G$, the cover $\{gU\}\_{g \in G}$ has a countable subcover, or equivalently every cover $\{Ug\}\_{g \in G}$ has a countable subcover. Thank you.



Is a weakly separable group always Lindelöf?
Let's take the group $G = \mathbb Z^T$, where $T$ is an uncountable set. 


Is $G$ weakly separable? Yes: If $U$ is any neighborhood of $0$ in $G$, then there exists a finite set $T\_0 \subset T$ so that
$$
U \supseteq V := \{\phi \in G \colon \phi(t)=0 \text{ for all } t \in T\_0\}
$$
Countably many translates of $V$ cover $G$: namely translates by all $\psi \in G$ supported by $T\_0$.


If $G$ Lindelof? No. Steen & Seebach, Example 103.


