Regularity is not refining-preserved

Property-theoretic statement
The property of topological spaces of being a regular space does not satisfy the metaproperty of topological spaces of being refining-preserved.

Verbal statement
It is possible to have a regular space such that choosing a finer topology yields a space that is not regular.

Related facts

 * Hausdorff not implies regular: The fact that regularity is not preserved on passing to finer topologies clearly shows that regularity cannot be the same as Hausdorffness, because Hausdorffness is refining-preserved. More explicitly, the examples we construct to show that regularity is not refining-preserved, also serve as examples of Hausdorff non-regular spaces.

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 regular space under the usual topology. 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, so this space is not a regular space.

Textbook references

 * , Page 197, Example 1, Chapter 4, Section 31 (Munkres is using this example to construct a Hausdorff non-regular space, and does not mention here that this also shows that passing to a finer topology fails to preserve regularity)