Shrinking certain sliced decompositions of 𝐸ⁿ⁺¹

Daverman, Robert J.; Preston, D. Kriss

doi:10.1090/S0002-9939-1980-0567997-8

Shrinking certain sliced decompositions of $E^{n+1}$
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by Robert J. Daverman and D. Kriss Preston PDF

Proc. Amer. Math. Soc. 79 (1980), 477-483 Request permission

Abstract:

We set forth a connection, based on relatively elementary techniques, between the shrinkability of product decompositions of ${E^{n + 1}}$ and that of sliced decompositions. In particular, if G is a decomposition of ${E^{n + 1}}$ such that each decomposition element g is contained in some horizontal slice ${E^n} \times \{ s\}$ and if the decomposition ${G^s}$ of ${E^n}$, consisting of those subsets g of ${E^n}$ for which $g \times \{ s\} \in G$ , expands to a shrinkable decomposition ${G^s} \times {E^1}$ of ${E^n} \times {E^1}$, we show then that G itself is shrinkable.

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