I just started reading about topology and semi-algebraic sets, so please forgive me if this is a naive question:
Given a bounded and closed semi-algebraic set, which topological property guarantees that it is possible to shrink the set's boundary towards any point within the set without intersecting holes? (My guess so far: connected and contractible set).
Also: are there practical ways to determine whether this property holds for a given bounded and closed semi-algebraic set?