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
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.

## Preview

Unable to display preview. Download preview PDF.

## Copyright information

© Springer Science+Business Media New York 2002