A SEPARABLE NORMAL TOPOLOGICAL GROUP WHICH IS NOT LINDELijF
نویسندگان
چکیده
In 1968 Wilansky asked whether a separable normal topological group must be Lindelof. In [7] this question was answered in the negative, assuming CH, by constructing a hereditarily separable, normal group which is not Lindeliif. We give an example of a separable normal group which contains a closed subspace homeomorphic to an uncountable regular cardinal, in ZFC only. Of course we have to sacrifice hereditary separability, since TodorEeviC (and a little later Baumgartner) showed that it is consistent to assume that hereditarily separable regular spaces are Lindeliif (in short: ‘There are no S-spaces’). For details see [lo]. In Section 2 we associate with each topological space X a group B(X), which has a topology such that all translations are continuous, and show that in some special cases B(X) is a topological group. In Section 3 we give a space X whose B(X) will be the desired example. Our B(X) construction was somewhat inspired by the work done concerning free topological groups. For details and references, see Smith-Thomas [ 1 I].
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