Linear homeomorphic classification of spaces of continuous functions defined on SA

For a subset Aof the real line R, modification of the Sorgenfrey line SAis a topological space whose underlying points set is the reals Rand whose topology id defined as follows: points from Aare given the neighbourhoods of the right arrow while remaining points are given the neighbourhoods of the Sorgenfrey line S(or left arrow). A necessary and sufficient condition under which the space Cp(SA)is linearly homeomorphic to Cp(S)is obtaine

On a homeomorphism between the Sorgenfrey line S and its modification Sp

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