Hausdorff not implies regular

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
View a complete list of topological space property non-implications | View a complete list of topological space property implications |Get help on looking up topological space property implications/non-implications
|

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