Lindelof x Metric Needs Not Be Normal

In a previous post, an example for Paracompact x Metric needs not be normal was given. The example is product of the Michael Line and the space of irrationals. In another post, I showed that as long as the set of isolated points in the “Michael Line” construction is not an F_\sigma-set, we can obtain a non-normal product. I present here another example using the same technique, but this time the set of isolated points is a Bernstein set. This produces a Lindelof space and the cross product of this Lindelof space with the Bernstein set with the usual topology is not normal.

As in the previous post, let H \subset \mathbb{R}. Define a new topology on the real line \mathbb{R} by using open sets of the form U \cup V where U is an open set in the usual topology and V \subset H. In this notation, the Michael Line is \mathbb{R}(\mathbb{P}) where \mathbb{P} is the space of all irrational numbers. I proved in the previous post that the space \mathbb{R}(H) \times H is normal if and only if the set H is an F_\sigma-set in the real line. Thus \mathbb{R}(\mathbb{P}) \times \mathbb{P} is not normal.

I consider the example \mathbb{R}(S) where S \subset \mathbb{R} and both S and its complement \mathbb{R}-S contain no uncountable compact subset of the line. Such a set is called a Bernstein set. The space \mathbb{R}(S) is Lindelof. Let \mathcal{W} be an open cover of \mathbb{R}-S consisting of open sets in the usual topology. Then \mathcal{W} has to cover all of the real line except for countably many points. Otherwise \mathbb{R}-\cup \mathcal{W} is an uncountable closed set, which contains an uncountable compact set that is contained in S. Thus any open cover of \mathbb{R}-S made up of usual open sets has to cover the real line except for countably many points. Now let \mathcal{U} be an open cover of \mathbb{R}(S). The usual open sets in \mathcal{U} that covers \mathbb{R}-S has a countable subcover. This countable subcover covers all of the real line except for countably many points. Thus \mathcal{U} has a countable subcover.

Since a Berstein set cannot be an F_\sigma-set, \mathbb{R}(S) \times S is not normal. Thus the cross product of a Lindelof space and a separable metric space needs not be normal. Is there a non-nonmal example of hereditarily Lindelof x separable metric space? If there is, it will not be using this “Michael Line” type construction. It is easy to see that \mathbb{R}(H) is hereditarily Lindelof if and only if H is countable.

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