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 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 . Define a new topology on the real line by using open sets of the form where is an open set in the usual topology and . In this notation, the Michael Line is where is the space of all irrational numbers. I proved in the previous post that the space is normal if and only if the set is an set in the real line. Thus is not normal.
I consider the example where and both and its complement contain no uncountable compact subset of the line. Such a set is called a Bernstein set. The space is Lindelof. Let be an open cover of consisting of open sets in the usual topology. Then has to cover all of the real line except for countably many points. Otherwise is an uncountable closed set, which contains an uncountable compact set that is contained in . Thus any open cover of made up of usual open sets has to cover the real line except for countably many points. Now let be an open cover of . The usual open sets in that covers has a countable subcover. This countable subcover covers all of the real line except for countably many points. Thus has a countable subcover.
Since a Berstein set cannot be an set, 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 is hereditarily Lindelof if and only if is countable.