On pointwise $\mathpzc{M}$-continuity of mappings
Abstract
Classical Levine's theorem [N. Levine: \textit{Semi-open sets and semi-continuity in topological spaces}, Amer. Math. Monthly
{\bf70} (1963), 36--41] asserts that for a semi-continuous mapping on a second countable topological space, the discontinuity points
form a 1st category set. There are two directions in~literature in which this result is generalized: by considering either multi-valued
mappings or mappings on some second noncountable spaces (for the latter, see for instance [T.~Neubrunn:
\textit{Quasi-continuity (topical survey)}, Real Anal. Exchange {\bf14} (1988/89), 259--306]). In this paper,
we offer yet another path, namely, the path of so-called $\mathpzc{M}$-spaces, essentially weaker than the topological ones.
Pointwise $\mathpzc{M}$-continuity of a mapping between two $\mathpzc{M}$-spaces is defined and characterized.
These characterizations are the basic tool for our generalization.
{\bf70} (1963), 36--41] asserts that for a semi-continuous mapping on a second countable topological space, the discontinuity points
form a 1st category set. There are two directions in~literature in which this result is generalized: by considering either multi-valued
mappings or mappings on some second noncountable spaces (for the latter, see for instance [T.~Neubrunn:
\textit{Quasi-continuity (topical survey)}, Real Anal. Exchange {\bf14} (1988/89), 259--306]). In this paper,
we offer yet another path, namely, the path of so-called $\mathpzc{M}$-spaces, essentially weaker than the topological ones.
Pointwise $\mathpzc{M}$-continuity of a mapping between two $\mathpzc{M}$-spaces is defined and characterized.
These characterizations are the basic tool for our generalization.
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PDFDOI: https://doi.org/10.2478/tatra.v52i0.176