Beginning Functional Analysis pp 16-31 | Cite as

# The Topology of Metric Spaces

Chapter

## Abstract

Let (for any choice of

*M, d*) be a metric space. Recall that the*r-ball centered at x*is the set$$
B_r (x) = \left\{ {y \in M\left| {d(x,y) < r} \right.} \right\}
$$

*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*).## Keywords

Hilbert Space Compact Subset Limit Point Open Cover Prove Theorem## Preview

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## Copyright information

© Springer Science+Business Media New York 2002