## Monotone decompositions of $\theta$-continua

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- by E. E. Grace PDF
- Trans. Amer. Math. Soc.
**275**(1983), 287-295 Request permission

## Abstract:

A $\theta$-*continuum*(${\theta _n}$-

*continuum*) is a compact, connected, metric space that is not separated into infinitely many (more than $n$) components by any subcontinuum. The following results are among those proved. The first generalizes earlier joint work with E. J. Vought for ${\theta _n}$-continua, and the second generalizes earlier work by Vought for ${\theta _1}$-continua. A $\theta$-continuum $X$ admits a monotone, upper semicontinuous decomposition $\mathcal {D}$ such that the elements of $\mathcal {D}$ have void interiors and the quotient space $X/\mathcal {D}$ is a finite graph, if and only if, for each nowhere dense subcontinuum $H$ of $X$, the continuum $T(H) = \{x \in X|$ if $K$ is a subcontinuum of $X$ and $x$ is in the interior of $K$, then $K \cap H \ne \emptyset \}$ is nowhere dense. Also, if $X$ satisfies this condition, then $X$ is in fact a ${\theta _n}$-continuum, for some natural number $n$, and, for each natural number $m$, $X$ is a ${\theta _m}$-continuum, if and only if $X/\mathcal {D}$ is a ${\theta _m}$-continuum.

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## Additional Information

- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**275**(1983), 287-295 - MSC: Primary 54F20; Secondary 54B15, 54C60, 54E45, 54F65
- DOI: https://doi.org/10.1090/S0002-9947-1983-0678350-8
- MathSciNet review: 678350