Hausdorff not implies regular

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.

Textbook references

 * , Page 197, Example 1, Chapter 4, Section 31