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.¹²

References

Nyikos, Peter J. (2001), "A history of the normal Moore space problem", Handbook of the History of General Topology, Hist. Topol., vol. 3, Dordrecht: Kluwer Academic Publishers, pp. 1179–1212, ISBN 978-0-7923-6970-7, MR 1900271
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.

[nyikos-1] Nyikos, Peter J. (2001), "A history of the normal Moore space problem", Handbook of the History of General Topology, Hist. Topol., vol. 3, Dordrecht: Kluwer Academic Publishers, pp. 1179–1212, ISBN 978-0-7923-6970-7, 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.

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