54A05-LimitPointsOfSequences.tex

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\pmtitle{limit points of sequences}
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\pmtype{Definition}
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\pmdefines{limit point of a sequence}
\pmdefines{limit point of the sequence}
\pmdefines{accumulation point of a sequence}
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In a topological space $X$, a point $x$ is a \emph{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 \emph{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
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