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Characterizations of Baire$ \sp \ast1$ functions in general settings

Author: Darwin E. Peek
Journal: Proc. Amer. Math. Soc. 95 (1985), 577-580
MSC: Primary 26A21; Secondary 54C30
MathSciNet review: 810167
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Abstract: Baire* 1 functions from $ \left[ {0,1} \right]$ to $ R$ were defined by R. J. O'Malley. For a general topological space $ X$, a function $ f:X \to R$ will be said to be Baire* 1 if and only if for every nonempty closed subset $ H$ of $ X$, there is an open set $ U$ such that $ U \cap H \ne \emptyset $ and $ f\left\vert H \right.$ is continuous on $ U$. Several characterizations of Baire* 1 functions are found by altering the well-known Baire 1 characterization: If $ H$ is a nonempty closed subset of the domain of $ f$, then $ f\left\vert H \right.$ has a point where $ f\left\vert H \right.$ is continuous. These conditions simply replace "closed subset of the preceding characterization with "subset", "countable subset" or "dense-in-itself subset". The relationships of these characterizations are examined with the domain of $ f$ being various spaces. The independence of these conditions from the discrete convergence condition described by Á. Császár and M. Laczkovich is discussed.

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Keywords: Baire 1, Baire* 1, discrete convergence, hereditarily separable
Article copyright: © Copyright 1985 American Mathematical Society

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