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Here is how my book defines accumulation point.

**Let S be a point set, and let y be a point which is not necessarily in S. We call y an accumulation point of S if in each neighborhood of y there is at least one point x which is in S and distinct from y.**

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DEFINITION:

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Definition for an open set from my book:

**A point set S is called open if for each point x_0 of S there is some neighborhood of x_0 which belongs entirely to S.**

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Definition for a neighborhood from my book:

**By a neighborhood of a point x_0 we mean the set of all points x such that -h<x-x_0<h, where h is some positive number.**

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I think I don't understand the definition of accumulation point because I can't understand what the definition is saying geometrically. What I mean is, say we have a candidate accumulation point, call it y. So we check in every neighborhood of y to see if there exists an x which is in the set of interest and it isn't equal to y. But isn't there an infinite amount of neighborhoods of y?

If there is an infinite amount of neighborhoods of y, how can you check each of them to see if there exists an x which is in the set of interest and it isn't equal to y?

I completely confused myself, any help would be appreciated.