Abstract. We prove that: (i) a pathwise connected, Hausdorff space which has a continuous selection is homeomorphic to one of the following four spaces: singleton, [0, 1), [0,1] or the long line L, (ii) a locally connected (Hausdorff) space which has a continuous selection must be orderable, and (iii) an infinite connected, Hausdorff space has exactly two continuous selections if and only if it is compact and orderable. We use these results to give various characterizations of intervals via continuous selections. For instance, (iv) a topological space X is homeomorphic to [0, 1] if (and only if) X is infinite, separable, connected, Hausdorff space and has exactly two continuous selections, and (v) a topological space X is homeomorphic to [0, 1) if (and only if) one of the following equivalent conditions holds: (a) X is infinite, Hausdorff, separable, pathwise connected and has exactly one continuous selection; (b) X is infinite, separable, locally connected and has exactly one continuous selection; (c) X is infinite, metric, locally connected and has exactly one continuous selection. Three examples are exhibited which demonstrate the necessity of various assumptions in our results.MSC Classification: Primary 54C65; Secondary 54B20, 54C60, 54D05, 54D30, 54F05, 54F15.
