A function space triple of a compact polyhedron into an open set in Euclidean space

Sakai, Katsuro

doi:10.1090/S0002-9939-1990-0991709-3

A function space triple of a compact polyhedron into an open set in Euclidean space
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Proc. Amer. Math. Soc. 108 (1990), 547-555 Request permission

Abstract:

Let $X$ be a non-zero dimensional compact Euclidean polyhedron and $Y$ an open set in Euclidean space ${{\mathbf {R}}^r}\left ( {r > 0} \right )$. The spaces of (continuous) maps, Lipschitz maps and PL maps from $X$ to $Y$ are denoted by $C\left ( {X,Y} \right )$, ${\text {LIP}}\left ( {X,Y} \right )$ and ${\text {PL}}\left ( {X,Y} \right )$, respectively. We prove that the triple \[ \left ( {C\left ( {X,Y} \right ),{\text {LIP}}\left ( {X,Y} \right ){\text {PL}}\left ( {X,Y} \right )} \right )\] is an $\left ( {s,\Sigma ,\sigma } \right )$-manifold triple, where $s = {\left ( { - 1,1} \right )^\omega }$, \[ \Sigma = \left \{ {x \in s|\sup \left | {x\left ( i \right )} \right | < 1} \right \}{\text {and}} \sigma = \left \{ {x \in s|x\left ( i \right ) = 0 {\text {except}} {\text {for}} {\text {finitely}} {\text {many}} i} \right \}.\].

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