In a general topological space are these properties equivalent ? If not, is there a property (e.g. first countability) that metric spaces possess which makes them equivalent there ? Here are the definitions as I understand them.
Bolzano-Weierstrass Property: every infinite sequence $(x_n)$ in X has an accumulation point in X, i.e. a point x such that every open set that contains x contains an infinite number of the points in $(x_n)$
Sequentially compactness: every infinite sequence in X has a convergent subsequence, i.e. converges to a limit, i.e. for every open set O in the topology there is N such that for all n > N then $x_n$ ∊ O (where $(x_n)$ are the points in the subsequence).