Hausdorff not implies regular

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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 need not satisfy the second topological space property
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Statement

Property-theoretic 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 refining-preserved

Examples

One example is the space \mathbb{R}_K 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 (a,b) \setminus K where K = \{ 1/n \mid n \in \mathbb{N} \}. This topology is clearly finer than the usual topology on the reals, and the reals form a Hausdorff space under the usual topology, so \mathbb{R}_K is also Hausdorff (passing to a finer topology preserves Hausdorffness).

However, \mathbb{R}_K is not a regular space. For instance, the closed subset K 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