An exercise involving non-normal spaces

A space is normal if any two disjoint closed subsets of the space can be separated by disjoint open sets. A space is pseudonormal if any two disjoint closed subsets of the space, one of which is countable, can be separated by disjoint open sets. In this post, we present an interesting exercise that deals with non-normal spaces:

    Take a space that is not normal. Then determine whether it is pseudonormal. You can supply your own examples or you can start with several non-normal spaces listed below. Once you have a list, determine which ones are psuedonormal and which ones are not.

To make the exercise more interesting, we propose that the focus is on spaces that are T_1 (i.e. singleton sets are closed) and regular. Since regular Lindelof spaces are normal, we will be certain that any non-normal (and regular) space is not Lindelof.

In the previous post called Pseudonormal spaces, we identify four spaces that are known to be non-normal. Three of these spaces are not normal because one countable closed set and another closed set cannot be separated, hence not pseudonormal (one is the Sorgenfrey plane and one is the Niemmytzkis’ plane). The fourth non-normal space is pseudonormal.

Here’s a list of several other non-normal spaces previously discussed in this blog.

  • The Tychonoff Plank.
  • The sigma-product of \omega_1 many copies of \omega_1+1.
  • The product space \omega^{\omega_1}.
  • The product of the Michael line and the space of irrationals.
  • The product of countably many copies of the Michael line.
  • The product of a Lindelof space and a Bernstein set.
  • The Pixley-Roy space \mathcal{F}[\mathbb{R}].
  • Mrowka space, defined on a maximal almost disjoint family of subsets of \omega.

Readers are welcome to submit other examples of non-normal spaces. Submit examples by entering a comment below. Submitted examples that are different from the ones listed above will be appended to this post.

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\copyright \ 2014 \text{ by Dan Ma}

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