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

 

Types of almost continuity


Authors: B. D. Garrett and K. R. Kellum
Journal: Proc. Amer. Math. Soc. 88 (1983), 555-559
MSC: Primary 54C08; Secondary 54F20, 54F25
MathSciNet review: 699433
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Abstract: The differences for properties of the range space $ f(X)$ are considered for functions $ f:X \to Y$, where $ f$ is almost continuous relative to $ X \times Z$, and $ f(X) \subset Z \subset Y$. It is shown that if $ f$ is allowed to be almost continuous relative to $ X \times Z$, where $ X$ is a Peano continuum and $ Z$ is a locally connected metric space, then $ f(X)$ can be any type of subcontinuum of $ Z$. This contrasts the known results for the case where $ Z = f(X)$ and almost continuity is relative to $ X \times f(X)$. Outer almost continuous retracts ($ f:X \to X$ is almost continuous relative to $ X \times X$) and inner almost continuous retracts ($ f:X \to X$ is almost continuous relative to $ X \times f(X))$) are defined. Properties of outer almost continuous retracts, including the existence of an outer almost continuous retract $ M$ of a fixed point space $ X$, where $ M$ does not have the fixed point property, are found.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0699433-8
PII: S 0002-9939(1983)0699433-8
Keywords: Almost continuous function, almost continuous retract, Peano continuum, fixed point property, pseudoarc, absolute retract, Hilbert cube
Article copyright: © Copyright 1983 American Mathematical Society