Let $X$ topological space and $f:X\to X$ a function. We say that $f(x)\to \ell$ when $x\to a$ if
(1) For all open $V\ni\ell$, there is an open $U\ni a$ s.t. $f(U)\subset V$.
If $X$ is second-countable (i.e. the topology has a countable basis), then (1) is equivalent to
(2) for all sequence $x_n\to a$, the sequence $f(x_n)\to \ell$.
Questions
Q1) Why this caracterization doesn't hold if $X$ is not second countable ? i.e. I guess that (1)$\implies $ (2) always hold, but the converse is not true.
Q2) If $f:X\to Y$, to have the equivalence (1)$\iff$(2) do we need $X$ second countable, $Y$ second countable or $X$ and $Y$ second countable ?
