Open, Closed and Dense Subsets

Summary

Open and closed subsets of a metric space are at the heart of this chapter. The concept of density has also been introduced. We have explained the notion of a metric topology. We have shown in detail how the topology of a metric subspace relates to that of an enveloping superspace and have determined the topology for conserving metrics on a finite product. We have demonstrated that ℝ with its usual metric is complete—in other words, universally closed—and have begun a related discussion about nests of closed subsets.