Characterizations of intervals via continuous selections

Abstract

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.

Selections generating new topologies

“…According to [10,Lemma 7.2], T f is also an order topology on X provided X is connected. Finally, by [12,Theorem 4 and Remark 16], T f coincides with the original topology on X provided X is connected and locally connected.…”

Selections and weak orderability

“…Building on work of G. Artico, U. Marconi, J. Pelant, L. Rotter and M. Tkachenko [1], S. García Ferreira and M. Sanchis [7] showed that a pseudocompact space X admits a continuous weak selection if and only if theČech-Stone compactification βX is orderable, and consequently, if and only if X is suborderable (or a GO-space). Improving on Michael's result stated above, T. Nogura and G. Shakhmatov proved in [16] that a locally connected space X admits a continuous weak selection if and only if it is orderable. Recently, V. Gutev and T. Nogura [11], in a very nice survey article on the selection problem, restated van Mill-Wattel's question and asked, in particular, whether a locally compact space admitting a continuous weak selection is weakly orderable.…”

Vietoris continuous selections and disconnectedness-like properties

“…However, in case of connected spaces, this is so. The following result is a partial case of [7, Lemmas 7.2 and 7.3] (see also [9,Lemma 10]). …”