The Topology of Metric Spaces

Abstract

Let (M, d) be a metric space. Recall that the r-ball centered at x is the set $$ B_r (x) = \left\{ {y \in M\left| {d(x,y) < r} \right.} \right\} $$ for any choice of x ∈ M and r > 0. These sets are most often called open balls, open disks, or open neighborhoods, and they are denoted by the above or by B(x, r), D r (x), D(x, r), N r (x), N(x, r), among other notations. A point x ∈ M is a limit point of a set E ⊆ M if every open ball B r (x) contains a point y ≠ x, y ∈ E. If x ∈ E and x is not a limit point of E, then x is an isolated point of E. E is closed if every limit point of E is in E. A point x is an interior point of E if there exists an r > 0 such that B r (x) ⊆ E. E is open if every point of E is an interior point. A collection of sets is called a cover of E if E is contained in the union of the sets in the collection. If each set in a cover of E is open, the cover is called an open cover of 2s. If the union of the sets in a subcollection of the cover still contains E, the subcollection is referred to as a subcover for E. E is compact if every open cover of E contains a finite subcover. E is sequentially compact if every sequence of E contains a convergent subsequence. E is dense in M if every point of M is a limit point of E. The closure of E, denoted by Ē, is E together with its limit points. The interior of E, denoted by E° or int (E), is the set of interior points of E. E is bounded if for each x ∈ E, there exists r > 0 such that E ⊆ B r (x).