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A subset      A A       of a topological space is said to be semi-open if there exists an open set      U U       such that      U ⊆  A ⊆  Cl ⁡  ( U ) U \subseteq A \subseteq \operatorname {Cl} (U)       where      Cl ⁡  ( U ) \operatorname {Cl} (U)       denotes the closure of      U U      . The class of semi-open sets of a given topological space      ( X , T ) (X,\mathcal {T})       is denoted      S .O . ( X , T ) {\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      T \mathcal {T}       and      T ∗  {\mathcal {T}^ \ast }       are two topologies for a set      X X       such that      S .O . ( X , T ) ⊆  S .O . ( X , T ∗  ) {\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) \subseteq {\text {S}}{\text {.O}}{\text {.}}(X,{\mathcal {T}^ \ast })      , then      T ⊆  T ∗  \mathcal {T} \subseteq {\mathcal {T}^ \ast }      . In a corollary to this theorem, Levine states that if      S .O . ( X , T ) = S .O . ( X , T ∗  ) {\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) = {\text {S}}{\text {.O}}{\text {.}}(X,{\mathcal {T}^ \ast })      , then      T = T ∗  \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.

A correction to the paper: ‘‘Semi-open sets and semi-continuity in topological spaces” (Amer. Math. Monthly {\bf 70} (1963), 36–41) by Norman Levine