Proper n-shape and the Freudenthal compactification

Abstract

The notions of $n$-shape for compact pairs and proper $n$-shape for locally compact spaces were introduced in [1] and [2], respectively. In this paper, strengthening $n$-shape of pairs, we define the notion of relative $n$-shape of compact pairs. By constructing a functor from the proper $n$-shape category to the relative $n$-shape category, we prove that for locally compact spaces $X$ and $Y$ with $\dim\leq n+1$, if $n$-${\rm Sh}_{p}(X)=n$-${\rm Sh}_{p}(Y)$ then $n$-${\rm Sh}(FX, EX)=n$-${\rm Sh}(FY,EY)$ rel. $(EX,EY)$ and $n$-${\rm Sh}(CX, \{\infty\})=n$-${\rm Sh}(CY, \{\infty\})$ rel. $(\{\infty\}, \{\infty\})$, where $FX$ is the Freudenthal compactification of $X$ with $EX$ the ends, and $CX=X\cup\{\infty\}$ is the one-point compactification of X. As corollaries, (1) if $X$ is connected $SUV^{n}$ and $\dim X\leq n+1$, then $FX\in UV^{n}$, (2) if $X$, $Y\subset\mu_{\infty}^{n+1}=\mu^{n+1}\backslash \{*\}$ are $Z$-sets and $n$-${\rm Sh}_{p}(X)=n$-${\rm Sh}_{p}(Y)$, then $\mu_{\infty}^{n+1}\backslash X$ is homeomorphic $(\approx)$ to $\mu_{\infty}^{n+1}\backslash Y$.