limit points of sequences

In a topological space $X$, a point $x$ is a limit point of the sequence $x_0,x_1,\mathrm{\dots }$ 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 $x_0,x_1,\mathrm{\dots }$ 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