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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
##### Keywords:
$$K_W$$-space; $$K_1$$-space
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