It is known that, over $\text{RCA}_0$, the Heine-Borel theorem is equivalent to $\text{WKL}_0$ and that the Bolzano-Weierstrass theorem is equivalent to $\text{ACA}_0$.
In general, a topological space $X$ is compact if and only if every net on $X$ has a convergent subnet. This theorem can be viewed as a generalization of either the Bolzano-Weierstrass theorem or the Heine-Borel theorem. Is the reverse mathematical strength of this statement known? Is there even a generally accepted formulation of this statement in the langague of second-order arithmetic?
No, not really. Unlike algebra, general topology is not well-accommodated by the framework of second-order arithmetic. James Hunter's Ph.D. thesis Higher-order reverse topology handles this by doubly extending the language of reverse mathematics, first by incorporating arbitrary finite types (a la Kohlenbach's $\mathsf{RCA}_0^\omega$) and second by incorporating urelements besides natural numbers (representing the points in the spaces under consideration). I recommend looking at Hunter's work to get a sense of the difficulties that arise in approaching topology from a reverse-mathematical perspective.