Hausdorff space not completely Hausdorff
such set is the infinite arithmetic progression of positive integers with difference and containing . The collection of all sets is a basis for a topology on . We will use a coarser topology induced by the following basis:
The collection is basis for a topology on
We first prove such collection is a basis. Suppose . By Euclid’s algorithm we have and and
besides, since and then so and are coprimes and . This concludes the proof that is indeed a basis for a topology on .
The topology on induced by is Hausdorff
Taking , we can find an integer such that and such that . A way to accomplish this is to take any multiple of greater than and add .
We conclude that and this means that becomes a Hausdorff space with the given topology.
Some properties of
We need to determine first some facts about . in order to take an example, consider first. Notice that if we had considered the former topology (where in , and didn’t have to be coprime) the complement of would have been which is open, and so would have been closed. In general, in the finer topology, all basic sets were both open and closed. However, this is not true in our coarser topology (for instance is not open).
The key fact to prove is not a completely Hausdorff space is: given any , then is a subset of .
Indeed, any basic open set containing is of the form with coprimes. This means . Now and have common terms if an only if for some integers . But that diophantine equation can be rewritten as
and it always has solutions because divides .
This also proves , because is not in but it is on the closure.
The topology on induced by is not completely Hausdorff
Let different positive integers. Since is a basis, for any two disjoint neighborhoods we can find basic sets and such that
But then is both a multiple of and so it must be in and . This means
and thus .
This proves the topology under consideration is not completely Hausdorff (under both usual meanings).
|Title||Hausdorff space not completely Hausdorff|
|Date of creation||2013-03-22 14:16:05|
|Last modified on||2013-03-22 14:16:05|
|Last modified by||drini (3)|
|Synonym||example of a Hausdorff space that is not completely Hausdorff|