Exercise 9, § 2

Let $X$ be a topological space, each point of which has a countable fundamental system of neighbourhoods.

a) $X$ is Hausdorff provided that every convergent sequence in $X$ has only one limit.

b) If $a$ is a cluster point of an infinite sequence $(x_n)$ of points of $X$, then there is an infinite subsequence of $(x_n)$ which converges to $a$.

c) Let $A$ be a non-empty subset of $X$, let $x_0$ be a point of $\overline{A}$, and let $f$ be a mapping of $A$ into a Hausdorff topological space $X'$. Then a point $a \in X'$ is a limit of $f$ at the point $x_0$, relative to $A$, provided that, for every sequence $(x_n)$ of points of $A$ which converges to $x_0$, the sequence $(f(x_n))$ converges to $a$.

d) With the notation of $c_j$, suppose also that every point of $X'$ has a countable fundamental system of neighbourhoods. If $a \in X'$ is a cluster point of $f$ at $x_0$ relative to $A$, then there is a sequence $(x_n)$ of points of $A$ which converges to $x_0$ and is such that the sequence $(f(x_n))$ converges to $a$.