Proof : is a union of basis elements , with and open sets in and respect. Since is compact (it is homeomorphic to ), only a finite number of such basis elements cover .
Define . The set is open and contains because each intersects by the previous remark.
We now claim that . Let be a point in . The point is in some and so . We also know that .
Therefore as desired.
- 1 J. Munkres, Topology (2nd edition), Prentice Hall, 1999.