TITLE: Sequences in Hausdorff spaces
QUESTION [5 upvotes]: As far as I see is the sequence concept in Hausdorff spaces well-defined in the sense that we have unique limits. Thus, I was wondering whether the definition of the closure of a set that contains all possible limit points of sequences and the definition of a continuous function as $$x_n \rightarrow x \Rightarrow f(x_n) \rightarrow f(x)$$ still hold here or whether they are somehow weaker/stronger concepts in this context?

REPLY [2 votes]: You need the notion of nets for this. For instance, we have that a sequence in $\ell_1(\mathbb{N})$ converges to $x$ in the weak topology if and only if it converges in the $||\cdot||_1$ topology to $x$ (not entirely trivial). Both are Hausdorff, but have different topologies.
EDIT: Very sorry. Big mistake corrected I hope.