Adherent point

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In mathematics, an adherent point (also closure point or point of closure or contact point)[1] of a subset A of a topological space X, is a point x in X such that every open set containing x contains at least one point of A. A point x is an adherent point for A if and only if x is in the closure of A.

This definition differs from that of a limit point, in that for a limit point it is required that every open set containing x contains at least one point of A different from x. Thus every limit point is an adherent point, but the converse is not true. An adherent point of A is either a limit point of A or an element of A (or both). An adherent point which is not a limit point is an isolated point.

Intuitively, having an open set A defined as the area within (but not including) some boundary, the adherent points of A are those of A including the boundary.

Examples[edit]

  • If S is a subset of R which is bounded above, then sup S is adherent to S.
  • A subset S of a metric space M contains all of its adherent points if, and only if, S is closed in M.
  • In the interval (a, b], a is an adherent point that is not in the interval, with usual topology of R.

Notes[edit]

  1. ^ Steen, p. 5; Lipschutz, p. 69; Adamson, p. 15.

References[edit]