# 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
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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.

## 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