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Homelimit points of sequences

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# 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.

## Mathematics Subject Classification

54A05*no label found*

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