Topological groups and convex sets homeomorphic to non-separable Hilbert spaces

Taras Banakh; Igor Zarichnyy

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Let X be an infinite, locally connected, locally compact separable metrizable space. The space C(X) of real-valued continuous functions defined on X with the compact-open topology is a separable Fréchet space, so it is homeomorphic to the psuedo-interior s = (−1, 1)ℕ of the Hilbert cube Q = [−1, 1]ℕ. In this paper, generalizing the Sakai-Uehara’s result to the non-compact case, we construct a natural compactification $\bar{C}$ (X) of C(X) such that the pair ( $\bar{C}$ (X), C(X)) is homeomorphic to (Q,...