G separable group, $\aleph_0 \leq \tau$. What is $l(X)$ and $\omega l(X) (\leq \tau)$? where $X \subseteq G$. And what is $\chi (G)$ (cardinal)?

Question

Happy Chinese new year!

I was reading (and translating) a Russian article "On the topological groups close to being Lindelöf". Where it is assumed G is a separable group and $\tau \geq \aleph_0$

Theorem 1. Let a topological group G satisfy one of the following conditions:

1. G is algebraically generated by a subset $X \subseteq G$, where $l(Х) \leq \tau$ ;
2. G is topologically generated by a subset $X \subseteq G$, where $l(Х) \leq \tau$ ;
3. $ωl(G) \leq \tau$. Then ...

I'd like to know the definition of l(X) or where can I find this notation and definitions?

Here is the original text

Here is another notation I'd like to know

What is the definition $\chi (G_1)$ or where can I find it?

This is what I found but I'm no sure if it applies to my case: Posible definition of χ($G_1$)?

And if this χ has the same definition?

I thank you in advance for your help.