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Note on function spaces with the topology of pointwise convergence. (English) Zbl 1035.54011
The paper contains two examples of function spaces \(C_p(X)\) of continuous real-valued functions together with the topology of pointwise convergence. The first example is the Cook continuum. The author proves that for any Cook continuum \(M\) the space \(C_p(M)^n\) is uniformly homeomorphic to \(C_p(M)^m\) if and only if \(n=m\). The second example provides a negative answer to Problem 45 of [A. V. Arkhangel’skij, Hvsek, M. (ed.) et al., Recent progress in general topology, Amsterdam: North Holland, 1–56 (1992; Zbl 0814.54009)].

54C35 Function spaces in general topology
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54G20 Counterexamples in general topology
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