I know that in metric spaces sequentially compact is equivalent to compact. I also know that compact and sequentially compact for general topologies there is no relation between them. But I wanted to know if there is the same implication for Hausdorff I-countable spaces. The only thing I have found is that for Hausdorff spaces with the B-W property implies countably compact.
Any books where I can research more on the subject?
The $\pi$-Base may be a good place to start your search. For example, this search reveals that all compact first-countable spaces are sequentially compact, while this search shows examples of first countable sequentially compact spaces that fail to be compact.