CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A Stone space is a compact, Hausdorff totally disconnected topological space.
Stone spaces are sometimes called profinite spaces, since they are precisely the spaces which are small cofiltered limits of finite discrete spaces, and moreover (as a consequence of Stone duality) the category of Stone spaces is equivalent to the category $pro(FinSet)$ of pro-objects in FinSet and finite sets sit $FinSet\hookrightarrow pro(FinSet)$ as finite discrete spaces. This is especially common when talking about profinite groups and related topics.
A standard textbook is
See also
See also