Completely regular not implies normal

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This article gives the statement and possibly, proof, of a non-implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., Completely regular space (?)) need not satisfy the second topological space property (i.e., Normal space (?))
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A completely regular space need not be a normal space.


Example of the Moore plane

Further information: Moore plane (also called Niemitzky plane, or Niemitzky-Moore plane or tangent disk topology)

The Moore plane or Niemitzky plane or tangent disk topology is defined as follows: as a set, it is the upper half-plane, along with the bounding real line. The topology is described by the following basis:

  • All open disks that lie completely inside the upper half-plane
  • For points which are on the bounding line, the union of such a point with an open disk tangent to the bounding line at that point

We have that: