From Wikipedia:
One of the most important properties of first-countable spaces is that given a subset $A$, a point $x$ lies in the closure of $A$ if and only if there exists a sequence $\{x_n\}$ in $A$ which converges to $x$.
I was wondering if the above quote is equivalent to that there is no isolated point in a first-countable space?
I was wondering when"$f$ is a function on a first-countable space" as in the following quotes, what the codomain of $f$ is?
This has consequences for limits and continuity.
In particular, if $f$ is a function on a first-countable space, then $f$ has a limit $L$ at the point $x$ if and only if for every sequence $x_n → x$, where $x_n ≠ x$ for all $n$, we have $f(x_n) → L$.
Also, if $f$ is a function on a first-countable space, then $f$ is continuous if and only if whenever $x_n → x$, then $f(x_n) → f(x)$.
Thanks and regards!