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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

There is no exactly $ k$-to-$ 1$ function from any continuum onto $ [0,1]$, or any dendrite, with only finitely many discontinuities


Author: Jo W. Heath
Journal: Trans. Amer. Math. Soc. 306 (1988), 293-305
MSC: Primary 54C10; Secondary 54F15, 54F50
MathSciNet review: 927692
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Abstract: Katsuura and Kellum recently proved [8] that any (exactly) $ k$-to$ 1$ function from $ [0,\,1]$ onto $ [0,\,1]$ must have infinitely many discontinuities, and they asked if the theorem remains true if the domain is any (compact metric) continuum. The result in this paper, that any (exactly) $ k$-to-$ 1$ function from a continuum onto any dendrite has finitely many discontinuities, answers their question in the affirmative.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0927692-1
PII: S 0002-9947(1988)0927692-1
Keywords: $ k$-to-$ 1$ function, $ k$-to-$ 1$ map
Article copyright: © Copyright 1988 American Mathematical Society