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Transactions of the American Mathematical Society

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General position properties satisfied by finite products of dendrites


Author: Philip L. Bowers
Journal: Trans. Amer. Math. Soc. 288 (1985), 739-753
MSC: Primary 54F50; Secondary 54C25, 54C35, 54F35
DOI: https://doi.org/10.1090/S0002-9947-1985-0776401-5
MathSciNet review: 776401
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Abstract: Let $ \bar A$ be a dendrite whose endpoints are dense and let $ A$ be the complement in $ \bar A$ of a dense $ \sigma $-compact collection of endpoints of $ \bar A$. This paper investigates various general position properties that finite products of $ \bar A$ and $ A$ possess. In particular, it is shown that (i) if $ X$ is an $ L{C^n}$-space that satisfies the disjoint $ n$-cells property, then $ X \times \bar A$ satisfies the disjoint $ (n + 1)$-cells property, (ii) $ {\bar A^n} \times [ - 1,1]$ is a compact $ (n + 1)$-dimensional $ {\text{AR}}$ that satisfies the disjoint $ n$-cells property, (iii) $ {\bar A^{n + 1}}$ is a compact $ (n + 1)$-dimensional $ {\text{AR}}$ that satisfies the stronger general position property that maps of $ n$-dimensional compacta into $ {\bar A^{n + 1}}$ are approximable by both $ Z$-maps and $ {Z_n}$-embeddings, and (iv) $ {A^{n + 1}}$ is a topologically complete $ (n + 1)$-dimensional $ {\text{AR}}$ that satisfies the discrete $ n$-cells property and as such, maps from topologically complete separable $ n$-dimensional spaces into $ {A^{n + 1}}$ are strongly approximable by closed $ {Z_n}$-embeddings.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0776401-5
Keywords: Disjoint $ n$-cells property, discrete $ n$-cells property, locally $ n$-connected in $ X$
Article copyright: © Copyright 1985 American Mathematical Society

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