Monotone decompositions of -continua

Authors:
E. E. Grace and Eldon J. Vought

Journal:
Trans. Amer. Math. Soc. **263** (1981), 261-270

MSC:
Primary 54F20; Secondary 54B15

MathSciNet review:
590423

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Abstract: We prove the following theorem for a compact, metric -continuum (i.e., a compact, connected, metric space that is not separated into more than components by any subcontinuum). The continuum admits a monotone, upper semicontinuous decomposition such that the elements of have void interiors and the quotient space is a finite graph, if and only if, for each nowhere dense subcontinuum of , the continuum if is a subcontinuum of and , then is nowhere dense. The elements of the decomposition are characterized in terms of the set function . An example is given showing that the condition that requires to have void interior for all is not strong enough to guarantee the decomposition.

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DOI:
https://doi.org/10.1090/S0002-9947-1981-0590423-5

Keywords:
-continuum,
monotone upper semicontinuous decomposition,
quotient space,
finite graph,
aposyndetic set function ,
compact metric continuum

Article copyright:
© Copyright 1981
American Mathematical Society