Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 26A21, 54C30

Retrieve articles in all journals with MSC: 26A21, 54C30

Additional Information

PII: S 0002-9939(1985)0810167-0
Keywords: Baire 1, Baire* 1, discrete convergence, hereditarily separable
Article copyright: © Copyright 1985 American Mathematical Society