On compacta which are \(\ell\)-equivalent to \(I^ n\) (Q1100758)

Spaces X and Y ar said to be \(\ell\)-equivalent if \(C_ p(X)\) is linearly isomorphic to \(C_ p(Y)\), where \(C_ p(X)\) stands for the space of all realvalued continuous functions on X with the topology of pointwise convergence. This paper is devoted to metrizable spaces which are \(\ell\)- equivalent to n-manifolds. The following results are proved: (1) Let X be an n-dimensional locally compact space and Y be the closure of the set of all points of X whose local dimensions are exactly n. If X is \(\ell\)- equivalent to an n-manifold, then the set of all points of Y at which Y is locally contractible is dense in Y. (2) If a curve X is \(\ell\)- equivalent to a finitely Suslinian compactum, then the set of all points of X at which X is locally connected is dense and has a non-empty interior in X. (3) Every dendrite with finite ramification points is \(\ell\)-equivalent to [0,1]. (4) Every continuum which is a one-to-one continuous image of [0,\(\infty)\) is \(\ell\)-equivalent to [0,1].