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

 

Continuity of certain connected functions and multifunctions


Author: Melvin R. Hagan
Journal: Proc. Amer. Math. Soc. 42 (1974), 295-301
MSC: Primary 54C60
MathSciNet review: 0326652
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Abstract: In this paper it is proved that if X is a 1st countable, locally connected, $ {T_1}$-space and Y is a $ \sigma $-coherent, sequentially compact $ {T_1}$-space, then any nonmingled connectedness preserving multifunction from X onto Y with closed point values and connected inverse point values is upper semicontinuous. It follows that any monotone, connected, single-valued function from X onto Y is continuous. Let X be as above and let Y be a sequentially compact $ {T_1}$-space with the property that if a descending sequence of connected sets has a nondegenerate intersection, then this intersection must contain at least three points. If f is a monotone connected single-valued function from X onto Y, then f is continuous. An example of a noncontinuous monotone connected function from a locally connected metric continuum onto an hereditarily locally connected metric continuum is given.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0326652-7
PII: S 0002-9939(1974)0326652-7
Keywords: Upper semicontinuous multifunction, connectedness preserving function, monotone function, locally connected, $ \sigma $-coherent, hereditarily locally connected continuum
Article copyright: © Copyright 1974 American Mathematical Society