Fundamental properties of ε-connected sets

In this paper, we establish and discuss basic properties of ε-connected sets embedded in an arbitrary complete space. These properties are used to establish that X is ε-connected for all ε>0 if, and only if, X is connected in the original sense of Cantor [Math. Ann. 23 (1884) 453]. Such sets are “virtually connected” in a finite precision machine and in some circumstances can be used as efficient estimators of connected geometric figures.