Abstract:
The question of when the spaces $C_p(X)$ and $C_p(Y)$ of continuous real-valued functions on $X$ and $Y$ in the topology of pointwise convergence are linearly homeomorphic ($X$ and $Y$ are then called $l$-equivalent) is studied. The concept of Euclidean-resolvable compactum is introduced; it greatly generalizes the concept of a polyhedron, and it is proved that every Euclidean-resolvable space of dimension $n\geqslant 1$ is $l$-equivalent to the Euclidean cube $I^n$. It is established that if the dimensions of noncompact $CW$-spaces of countable weight coincide, then these spaces are $l$-equivalent. A complete zero-dimensional non-$\sigma$-compact metric space is $l$-equivalent to the space of irrational numbers. An elementary geometric technique based on factorizations is developed, making it possible to demonstrate $l$-equivalence.
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