A note on the weak topology of spaces \(C_k(X)\) of continuous functions
Let \(X\) be a Tychonoff space, and let \(C_k(X)\) and \(C_p(X)\) denote the spaces of continuous real-valued functions on \(X\) with the compact-open topology and the topology of pointwise convergence, respectively. Endow these spaces with the weak topology \(w\). The authors study the question of when the spaces \(C_k(X)_{w}\) and \(C_k(Y)_{w}\) are homeomorphic. Recall that a subset \(B\) of a locally convex space \(E\) is bounded if for each neighbourhood \(U\) of zero in \(E\) there is \(\lambda > 0\) with \(\lambda B \subset U\). Let \(\Delta\) be a hereditary topological property. Amongst others, the following is proved: (1) if \(X\) and \(Y\) are infinite Tychonoff spaces, then: (1.1) if \(C_p(X)\) and \(C_k(X)_{w}\) are homeomorphic and each bounded subset of \(X\) has property \(\Delta\), then each bounded subset of \(C_k(X)_{w}\) also has \(\Delta\); (1.2) if \(X\) is pseudocompact, \(Y\) is paracompact and locally compact and \(C_k(X)_w\) and \(C_k(Y)_w\) are homeomorphic and each bounded set in \(C_k(X)_w\) has property \(\Delta\), then each bounded set in \(C_k(Y)_w\) also has \(\Delta\); (2) if \(X\) is a Tychonoff space, then \(C_k(X)\) does not contain an isomorphic copy of \(\ell_1\) if and only if each compact subset of \(X\) is scattered; (3) let \(E\) be a Banach space and \(F\) a Fréchet locally convex space such that \(E_w\) and \(F_w\) are homeomorphic. If \(E\) does not contain any isomorphic copy of \(\ell_1\), then \(F\) has the same property.
scientific article; zbMATH DE number 7365223
scientific article