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 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 where . This topology is clearly finer than the usual topology on the reals, and the reals form a regular space under the usual topology. However, is not a regular space.

For instance, the closed subset 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)