General topology

Adherent point

In mathematics, an adherent point (also closure point or point of closure or contact point) of a subset of a topological space is a point in such that every neighbourhood of (or equivalently, every open neighborhood of ) contains at least one point of A point is an adherent point for if and only if is in the closure of thus if and only if for all open subsets if This definition differs from that of a limit point of a set, in that for a limit point it is required that every neighborhood of contains at least one point of different from Thus every limit point is an adherent point, but the converse is not true. An adherent point of is either a limit point of or an element of (or both). An adherent point which is not a limit point is an isolated point. Intuitively, having an open set defined as the area within (but not including) some boundary, the adherent points of are those of including the boundary. (Wikipedia).

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Related pages

Neighbourhood (mathematics) | Subset | Topological space | Metric space | Interval (mathematics) | Limit (mathematics) | Closure (topology) | Mathematics | Sequential space | Subspace topology | Isolated point | Empty set | Tom M. Apostol | Closed set