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
View a complete list of topological space property non-implications | View a complete list of topological space property implications |Get help on looking up topological space property implications/non-implications
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 Hausdorff space under the usual topology, so is also Hausdorff (passing to a finer topology preserves Hausdorffness).
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.
- Topology (2nd edition) by James R. Munkres, More info, Page 197, Example 1, Chapter 4, Section 31