In Elements of the Theory of Functions and Functional Analysis (Kolmogorov and Fomin) the definitions are as follows: An open sphere [itex]S(x_0,r)[/itex] in a metric space [itex]R[/itex] (with metric function [itex]\rho(x,y)[/itex]) is the set of all points [itex]x\in R[/itex] satisfying [itex]\rho(x,x_0)<r[/itex]. The fixed point [itex]x_0[/itex] is called the center; the number [itex]r[/itex] is called the radius. An ε-neighbourhood of the point [itex]x[/itex], denoted [itex]O(x,\epsilon)[/itex], is an open sphere of radius ε and center [itex]x_0[/itex]. How is the ε-neighbourhood a significant definition? It seems to be just the open sphere with a different radius symbol. If we have a neighbourhood of a point [itex]x[/itex] (as per the definition) does this [itex]x[/itex] have to lie within the open sphere? I mean, I assume it must, but this doesn't seem to be captured by the definition. What am I missing? Thanks.