Formula omitted]-Homogeneous continua with cut points

Abstract

A space is said to be 1 2 - homogeneous provided that there are exactly two orbits for the action of the group of homeomorphisms of the space onto itself. It is shown that if X is a 1 2 -homogeneous continuum with at least one cut point, then X has either uncountably many cut points or only one cut point c. In the former case, X is 1 2 -homogeneous if and only if X is an arc or X is a compactification of the reals R 1 whose remainder is the union of two disjoint, nondegenerate, homeomorphic homogeneous continua and the ends of X are mutually homeomorphic and 1 3 -homogeneous. In the latter case, the closures of the components of X − { c } are mutually homeomorphic and 2-homogeneous at c, and ord c ( X ) ⩾ 4 ; furthermore, if ord c ( X ) ⩽ ω , X is a locally connected bouquet of simple closed curves. Conversely, the two conditions about the components of X − { c } are shown to imply X is 1 2 -homogeneous under an additional assumption, which is shown by examples to be both required and restrictive.