General position properties satisfied by finite products of dendrites

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.