A Correction to the Paper "Semi-Open Sets and Semicontinuity in Topological Spaces" by Norman Levine

Abstract

A subset A of a topological space is said to be semi-open if there exists an open set U such that U C A C Cl(U) where Cl(U) denotes the closure of U. The class of semi-open sets of a given topological space (X, J) is denoted S.O. (X, J). In the paper Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41, Norman Levine states in Theorem 10 that if J and V* are two topologies for a set X such that S.O.(X, 3) C S.O.(X, J*), then 'J C P. In a corollary to this theorem, Levine states that if S.O.(X, if) = S.O.(X,Jf*), then _T= f*. An example is given which shows the above-mentioned theorem and its corollary are false. This paper shows how different topologies on a set which determine the same class of semi-open subse,ts can arise from functions, and points out some of the implications of two topologies being related in this manner. In [6] Norman Levine defines a set A in a topological space X to be semi-open if there exists an open set U such that U C A C Cl (U), where Cl(U) denotes the closure of U. The class of semi-open sets for a given topological space (X, i) is denoted S.O. (X, sT). Levine states in Theorem 10 of [6] that if Jf and * are two topologies for a set X such that S.O. (X, i) C S.O. (X, J9 then iT C *. In a corollary to this theorem, Levine states that if S.O. (X, ) = S.O. (X, iT*), then Jf = - The following example which is due to S. Gene Crossley and S. K. Hildebrand [1, Example 1.1] shows the above-mentioned theorem and its corollary are false. Example. Let X = la, b, cl, J; = t0, sal, la, bl, la, cl, XI, ;* = 10, sal, la, bl, XI. An exhaustion of all possibilities shows that S.O. (X, ) = S.O. (x, 5j*). Crossley and Hildebrand [3] defined two topologies iT and T* on a set X to be semi-correspondent if S.O. (X, J) = S.O. (X, 5f*). It is shown in [3] that semi-correspondence is an equivalence relation on the collection of Received by the editors March 3, 1974. AMS (MOS) subject classifications (1970). Primary 54B99; Secondary 54C10.