We establish that an uncountable space X must be essentially uncountable whenever its extent and tightness are countable. As a consequence, the equality $$\mathrm{ext}(X)= t(X)=\omega $$ implies that the space $$C_{p}(X, [0,1])$$ is discretely selective. If X is a metrizable space, then $$C_{p}(X, [0,1])$$ has the Banakh property if and only if so does $$C_{p}(Y, [0,1])$$ for some closed separable $$Y\subset X$$. We apply the above results to show that, for a metrizable X, the space $$C_{p}(X, [0,1])$$ is strongly dominated by a second countable space if and only if X is homeomorphic to $$D\,{\oplus }\, M$$ where D is a discrete space and M is countable. For a metrizable space X, we also prove that $$C_{p}(X,[0,1])$$ has the Lindelof $$\Sigma $$-property if and only if the set of non-isolated points of X is second countable. Our results solve several open questions.