Pseudonormal space

In mathematics, in the field of topology, a topological space is said to be pseudonormal if given two disjoint closed sets in it, one of which is countable, there are disjoint open sets containing them.^[1] Note the following:

• Every normal space is pseudonormal.
• Every pseudonormal space is regular.

An example of a pseudonormal Moore space that is not metrizable was given by F. B. Jones (1937), in connection with the conjecture that all normal Moore spaces are metrizable.^[1][2]

References

1. Nyikos, Peter J. (2001), "A history of the normal Moore space problem", Handbook of the History of General Topology, Hist. Topol., 3, Dordrecht: Kluwer Academic Publishers, pp. 1179–1212, MR 1900271
2. Jones, F. B. (1937), "Concerning normal and completely normal spaces", Bulletin of the American Mathematical Society, 43 (10): 671–677, doi:10.1090/S0002-9904-1937-06622-5, MR 1563615.