A correction to the paper: “Semi-open sets and semi-continuity in topological spaces” (Amer. Math. Monthly 70 (1963), 36–41) 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 $U \subseteq A \subseteq \operatorname {Cl} (U)$ where $\operatorname {Cl} (U)$ denotes the closure of $U$. The class of semi-open sets of a given topological space $(X,\mathcal {T})$ is denoted ${\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T})$. 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 $\mathcal {T}$ and ${\mathcal {T}^ \ast }$ are two topologies for a set $X$ such that ${\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) \subseteq {\text {S}}{\text {.O}}{\text {.}}(X,{\mathcal {T}^ \ast })$, then $\mathcal {T} \subseteq {\mathcal {T}^ \ast }$. In a corollary to this theorem, Levine states that if ${\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) = {\text {S}}{\text {.O}}{\text {.}}(X,{\mathcal {T}^ \ast })$, then $\mathcal {T} = {\mathcal {T}^ \ast }$. 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 subsets can arise from functions, and points out some of the implications of two topologies being related in this manner.