Abstract
For a compact space K we denote by $$C_w(K)$$ ( $$C_p(K)$$ ) the space of continuous real-valued functions on K endowed with the weak (pointwise) topology. In this paper we address the following basic question which seems to be open: Suppose that K is an infinite (metrizable) compact space. Can $$C_w(K)$$ and $$C_p(K)$$ be homeomorphic? We show that the answer is “no”, provided K is an infinite compact metrizable C-space. In particular our proof works for any infinite compact metrizable finite-dimensional space K.
Highlights
Mathematics Subject Classification 46E10 · 54C35 For a compact space K we can consider three natural topologies on the set C(K ) of all continuous real-valued functions on K : the norm topology, the weak topology and the pointwise topology
There is a vast literature studying the weak and the pointwise topology in function spaces, but surprisingly it seems to be unknown whether these two topologies are homeomorphic
In this note we show that Cw(K ) and C p(K ) are not homeomorphic for any infinite compact metrizable C-space K, in particular, for any infinite finitedimensional compact metrizable space K
Summary
Where F ∈ [K ]
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