Regularity is not refining-preserved

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This article gives the statement, and possibly proof, of a topological space property not satisfying a topological space metaproperty .
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Statement

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.

References

Textbook references

  • Topology (2nd edition) by James R. MunkresMore info, 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)