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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A correction to the paper: ``Semi-open sets and semi-continuity in topological spaces'' (Amer. Math. Monthly 70 (1963), 36-41) by Norman Levine


Author: T. R. Hamlett
Journal: Proc. Amer. Math. Soc. 49 (1975), 458-460
MSC: Primary 54A10
MathSciNet review: 0367888
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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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0367888-X
PII: S 0002-9939(1975)0367888-X
Keywords: Semi-continuous, semi-correspondent, semi-open
Article copyright: © Copyright 1975 American Mathematical Society