Appendix A - Preliminary material

Summary

Only separable metrizable spaces will be considered. Appendix A will be used to gather various notions and facts that are found in well–known reference books (for example, K. Kuratowski [85]) with the goal of setting consistent notation and easing the citing of facts. Also, the final Section A.7 will be used to present a proof, which includes a strengthening due to R. B. Darst [37], of a theorem of R. Purves [129].

A.1. Complete metric spaces

A metric that yields the topology for a metrizable space need not be complete. But this metric space can be densely metrically embedded into another metric space that is complete. A space that possesses a complete metric will be called completely metrizable. It is well–known that a G_δ subspace of a completely metrizable space is also completely metrizable (see J. M. Aarts and T. Nishiura [1, page 29]). Consequently,

Theorem A.1. A separable metrizable space is completely metrizable if and only if it is homeomorphic to a G_δ subset of a separable completely metrizable space.

The collection of all separable metrizable spaces will be denoted by MET and the collection of all completely metrizable spaces in MET will be denoted by MET_comp.

Extension of a homeomorphism. In the development of absolute notions it is often useful to extend a homeomorphism between two subsets of completely metrizable spaces to a homeomorphism between some pair of _δ subsets containing the original subsets. This is accomplished by means of the M. Lavrentieff theorem [89].