Given a nonstandard extension $X\hookrightarrow {}^\ast \!X$ (with the star operator defined for all subsets), by the first item on your list, knowledge of the halos allows you to recapture the notion of an open set. Namely, $S$ is open iff $\mu(s) \subseteq {}^*S$ for all $s \in S$. You solved your own problem then.

Given a nonstandard extension $X\hookrightarrow {}^\ast \!X$ (with the star operator defined for all subsets), by the first item on your list, knowledge of the halos allows you to recapture the notion of an open set. Namely, $S$ is open iff $\mu(s) \subseteq {}^*S$ for all $s \in S$. You solved your own problem then.

The concept that you mentioned such as continuity, compactness, etc., are precisely why we need a topology to define them in the traditional approach. Since continuity, compactness, etc. can now be defined without open sets, your problem has been solved.

One of my favorite examples is the one-line proof of Cantor's intersection theorem via halos.