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Endohomeomorphisms Decomposing a Space into Disjoint Copies of a Subspace

Liam O'Callaghan
Proceedings of the American Mathematical Society
Vol. 72, No. 2 (Nov., 1978), pp. 391-396
DOI: 10.2307/2042813
Stable URL: http://www.jstor.org/stable/2042813
Page Count: 6
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Endohomeomorphisms Decomposing a Space into Disjoint Copies of a Subspace
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Abstract

The existence (conjectured by R. Levy in a private communication) of a space $X$ and an endohomeomorphism, $f$, of $\beta X$, such that $f\lbrack X \rbrack = \beta X\backslash X$ is demonstrated. It is shown that if $G$ is one of the topological groups $\mathbf{2}^\alpha, \mathbf{Q}^\alpha, \mathbf{R}^\alpha$ or $\mathbf{T}^\alpha$, where $\omega < \alpha$, then $G$ has a dense $C$-embedded subgroup $H$ and an autohomeomorphism, $f$, such that $G$ is the union of disjoint sets, $A_0$ and $A_1$, where for $\{ i, j\}= \{0, 1\}f\lbrack A_i \rbrack = A_j$, and $A_i$ is a union of cosets of $H$.

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