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\(K_W\) does not imply \(K_W^*\). (English) Zbl 0963.54005
Summary: We prove that the cyclic monotonically normal space \(T\) of M. E. Rudin is a \(K_W\)-space which is not a \(K^*_W\)-space. This answers a question in [the author, ibid. 51, No. 1, 109-117 (1994; Zbl 0861.54012)]. In order to do this, we first prove that if a space \(X\) has \(D^*(\mathbb{R};\leq)\) then \(X\) is a \(K_W\)-space (it is well known that \(X\) is also a \(K_1\)-space; this does not necessarily mean that \(X\) is a \(K_{1W}\)-space).
MSC:
54C30 Real-valued functions in general topology
54C20 Extension of maps
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