Hausdorff not implies regular
This article gives the statement and possibly, proof, of a nonimplication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property need not satisfy the second topological space property
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Contents
Statement
Propertytheoretic statement
The property of topological spaces of being a Hausdorff space does not imply the property of being a regular space.
Verbal statement
There exist Hausdorff spaces that are not regular.
Related facts
Regularity is not refiningpreserved
Examples
One example is the space constructed by Munkres. The underlying set is the reals, and the basis is chosen as the usual open intervals, along with all sets of the form where . This topology is clearly finer than the usual topology on the reals, and the reals form a Hausdorff space under the usual topology, so is also Hausdorff (passing to a finer topology preserves Hausdorffness).
However, is not a regular space. For instance, the closed subset in this space (closed because its complement is open by construction in this topology) and the point 0 cannot be separated by disjoint open subsets.
References
Textbook references
 Topology (2nd edition) by James R. Munkres, ^{More info}, Page 197, Example 1, Chapter 4, Section 31