Note on function spaces with the topology of pointwise convergence
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The note contains two examples of function spaces C p (X) endowed with the pointwise topology. The first example is C p (M), M being a planar continuum, such that C p (M) m is uniformly homeomorphic to C p (M) n if and only if m = n. This strengthens earlier results concerning linear homeomorphisms. The second example is a non-Lindelöf function space C p (X), where X is a monolithic perfectly normal compact space all linearly orderable closed subspaces of which are metrizable. This example is obtained under the additional set-theoretical axiom ◊. This solves a problem of Arhangel’skiĭ.
Mathematics Subject Classification (1991):54C35
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