MathGloss

A subset EE of the metric space (X,ρ)(X,\rho) is totally bounded if for all ε>0\varepsilon > 0, there exists a finite set xii=1n{x_i}_{i=1}^n of points in XX such that Ei=1nB(xi,ε),E \subseteq \bigcup_{i=1}^n B(x_i, \varepsilon), where B(xi,ε)=xXρ(xi,x)<ε.B(x_i, \varepsilon) = {x \in X \mid \rho(x_i,x)< \varepsilon}. That is, for every ε>0\varepsilon > 0, EE can be covered by a finite set of balls of radius ε\varepsilon.

Wikidata ID: Q1362228