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.{{citation

| last = Nyikos | first = Peter J.

| contribution = A history of the normal Moore space problem

| mr = 1900271

| pages = 1179–1212

| publisher = Kluwer Academic Publishers | location = Dordrecht

| series = Hist. Topol.

| title = Handbook of the History of General Topology | url = https://books.google.com/books?id=dV6WtepcZLkC&pg=PA1184

| volume = 3

| year = 2001| isbn = 978-0-7923-6970-7

}} 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 {{harvs|first=F. B.|last=Jones|authorlink=F. Burton Jones|year=1937|txt}}, in connection with the conjecture that all normal Moore spaces are metrizable.{{citation

| last = Jones | first = F. B. | authorlink = F. Burton Jones

| doi = 10.1090/S0002-9904-1937-06622-5

| issue = 10

| journal = Bulletin of the American Mathematical Society

| mr = 1563615

| pages = 671–677

| title = Concerning normal and completely normal spaces

| volume = 43

| year = 1937| doi-access = free

}}.