My attempt to prove the statement on the title: the basis for some topology on $X$ can be finite or infinite.
If the basis is finite then exist some countable infinite subset that belongs only to an unique open set, so the induced topology is homeomorphic to $(\Bbb N,\tau)$ with $\tau$ indiscrete.
If the basis is infinite then exist some countable infinite subset where each element belongs to a different open set of the basis so the induced subspace is homeomorphic to $(\Bbb N,\tau)$ with $\tau$ being $T_0$.
It is my proof correct? It lacks something?
There is a different way to prove this? Can you outline to me some different strategy?
If my proof is fine, there is a way to express better the obvious statement that, for example, if the basis is finite and X is infinite then some infinite subspace is indiscrete?
Thank you in advance.