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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
DOI: https://doi.org/10.1090/S0002-9947-1988-0927692-1
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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Keywords: <IMG WIDTH="17" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img14.gif" ALT="$k$">-to-<IMG WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img15.gif" ALT="$1$"> function, <IMG WIDTH="17" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$k$">-to-<IMG WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$1$"> map
Article copyright: © Copyright 1988 American Mathematical Society