# limit points of sequences

In a topological space $X$, a point $x$ is a limit point of the sequence $x_{0},x_{1},\ldots$ if, for every open set containing $x$, there are finitely many indices such that the corresponding elements of the sequence do not belong to the open set.

A point $x$ is an accumulation point of the sequence $x_{0},x_{1},\ldots$ if, for every open set containing $x$, there are infinitely many indices such that the corresponding elements of the sequence belong to the open set.

It is worth noting that the set of limit points of a sequence can differ from the set of limit points of the set of elements of the sequence. Likewise the set of accumulation points of a sequence can differ from the set of accumulation points of the set of elements of the sequence.

Reference: L. A. Steen and J. A. Seebach, Jr. “Counterxamples in Topology” Dover Publishing 1970

Title limit points of sequences LimitPointsOfSequences 2013-03-22 14:38:13 2013-03-22 14:38:13 rspuzio (6075) rspuzio (6075) 7 rspuzio (6075) Definition msc 54A05 limit point of a sequence limit point of the sequence accumulation point of a sequence accumulation point of the sequence