I am taking a course in Algebraic Topology, and we have covered extension problems - that is, questions of the form "Does $f : X \to Y$ extend continuously to a map $\hat{f} : \hat{X} \to Y$ such that $\hat{f}|_{X} = f$ ?".
I was pondering this type of question and came up with this characterisation of a sequence having a limit:
Let $X$ be a Hausdorff space and $\mathbb{N} \stackrel{f}{\to} X : n \mapsto x_n$ a sequence in $X$. Then $(x_n)$ converges if and only if $f$ extends continuously to a map $\hat{f} : \mathbb{N}^* \to X$. Furthermore, if this holds then the limit of $(x_n)$ is $\hat{f}(\infty)$.
(Here $\mathbb{N}^* = \mathbb{N} \cup \{\infty\}$ is the one-point compactification of the naturals.)
My question is if this is actually a characterisation of a sequence having a limit, or if I have missed/assumed something that makes this not work. Also, do generalisations of this to spaces other than $\mathbb{N}$ have any interesting properties?