sequential continuity vs. continuity A short and hopefully simple question for someone with more experience in topology: 

If a topology is induced by a mode of convergence and in fact nothing more is known apriori, whether this topology is metrizable, first-countable or anything else? Is continuity then equivalent to sequential continuity? 

I'm quite sure this is should be true, but topology is tricky and my intuition not too developed in this field.

Let be X and Y topological spaces and $f : X \rightarrow Y$ is a map.
Then it holds:
1) If $f$ is continuous then it is sequentially continuous.
2) If $f$ is sequentially continuous and $X$ is first countable, then $f$ is continuous.
The prove of 1) is easy. You just have to use the definition.
For the prove of 2) you have to use that: $f$ is continuous => $f(\overline{M}) \subseteq \overline{f(M)}$
So there is equivalence if and only if $X$ is first countable.
