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A correction to the paper: 'Semi-open sets and semi-continuity in topological spaces' (Amer. Math. Monthly 70 (1963), 36–41) by Norman Levine
- Source :
- Proceedings of the American Mathematical Society. 49:458-460
- Publication Year :
- 1975
- Publisher :
- American Mathematical Society (AMS), 1975.
-
Abstract
- 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.
Details
- ISSN :
- 10886826 and 00029939
- Volume :
- 49
- Database :
- OpenAIRE
- Journal :
- Proceedings of the American Mathematical Society
- Accession number :
- edsair.doi...........fb8cf00a9181f07fc8b7b75e392b0746