 Indeed, the study of these properties even among subsets of Euclidean space, and the recognition of their independence from the particular form of the Euclidean metric, played a large role in clarifying the notion of a topological property and thus a topological space.However, whereas the structure of "compact" subsets of Euclidean space was understood quite early on via the Heine–Borel theorem, "connected" subsets of formula_1 (for "n" > 1) proved to be much more complicated.Indeed, while any compact Hausdorff space is locally compact, a connected space—and even a connected subset of the Euclidean plane—need not be locally connected (see below).