On a Homeomorphism between the Sorgenfrey Line S and Its Modification SP

Abstract

A topological space S_P, which is a modification of the Sorgenfrey line S, is considered. It is defined as follows: if x ∈ P ⊂ S, then a base of neighborhoods of x is the family {[x, x + ε), ε > 0} of half-open intervals, and if x ∈ SP, then a base of neighborhoods of x is the family {(x − ε, x], ε > 0}. A necessary and sufficient condition under which the space S_P is homeomorphic to S is obtained. Similar questions were considered by V. A. Chatyrko and I. Hattori, who defined the neighborhoods of x ∈ P to be the same as in the natural topology of the real line.