An epsilon-neighborhood of a point ( x ) in a metric space is the set of all points within a distance ( epsilon ) from ( x ), often denoted as ( B_epsilon(x) = { y mid d(x, y) < epsilon } ), where ( d ) is the distance function. This concept is fundamental in topology and analysis because it helps in defining continuity, limits, and open sets by capturing the idea of points being arbitrarily close to each other.