Clearly 1 implies 2 (by setting and ). To see that 2 implies 1, first take the transformation
so that . Then take the transformation , where and for all . Then and . Here, and denote the binary operations of taking the maximum and minimum of two given real numbers (see ring of continuous functions for more detail). Since during each transformation, the resulting function remains continuous, the first assertion is proved. ∎
Definition. Any two sets in a topological space satisfying the above equivalent conditions are said to be completely separated. When and are completely separated, we also say that is completely separated.
Remark. A T1 topological space in which every pair of disjoint closed sets are completely separated is a normal space. A T0 topological space in which every pair consisting of a closed set and a singleton is completely separated is a completely regular space.
|Date of creation||2013-03-22 16:54:51|
|Last modified on||2013-03-22 16:54:51|
|Last modified by||CWoo (3771)|