Sorgenfrey plane

In topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures.

The Sorgenfrey line and plane are named for the American mathematician Robert Sorgenfrey.

A basis for the Sorgenfrey plane, denoted

from now on, is therefore the set of rectangles that include the west edge, southwest corner, and south edge, and omit the southeast corner, east edge, northeast corner, north edge, and northwest corner.

is an uncountable discrete subset of this space, and this is a non-separable subset of the separable space

It shows that separability does not inherit to closed subspaces.