The Topology of Metric Spaces

  • Karen Saxe
Part of the Undergraduate Texts in Mathematics book series (UTM)

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 yx, 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 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).

Keywords

Hilbert Space Compact Subset Limit Point Open Cover Prove Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Karen Saxe
    • 1
  1. 1.Mathematics DepartmentMacalester CollegeSt. PaulUSA

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