It can be shown that a topological space is uniformizable iff it is completely regular.
Clearly, every pseudometric space is uniformizable. The converse is true if the space has a countable basis. Pushing this idea further, one can show that a uniformizable space is metrizable iff it is separating (or Hausdorff) and has a countable basis.
Let , , and be defined as above. Then is said to be completely uniformizable if is a complete uniformity.
Every paracompact space is completely uniformizable. Every completely uniformizable space is completely regular, and hence uniformizable.
|Date of creation||2013-03-22 16:49:05|
|Last modified on||2013-03-22 16:49:05|
|Last modified by||CWoo (3771)|