Definition of accumulation points

Question

Suppose $X$ is a metric space, $z \in X$ and $S \subset X$.

Then $z$ is called an accumulation point of $S$ in $X$ if, and only if,

dist(z,S\z) = 0.

But such points need not be members of $S$, right?

So i can understand the above given definition if $z \in S$ but if it doesn't belong to $S$ then,

We will probably have to alter the definition by saying that the dist(z,S) = 0, right ?

I am not sure. Am i correct ?

If $z \notin S$, then $S\setminus{z} = S$, no problem. – Daniel Fischer Oct 09 '13 at 09:31 — Daniel Fischer, Oct 09 '13 at 09:31

score 2 · Accepted Answer · answered Oct 09 '13 at 14:53

We will probably have to alter the definition by saying that the dist(z,S) = 0, right ?

We could but we don't have to, because $S$ and $S\setminus \{z\}$ are the same thing when $z$ does not belong to $S$. (As D.F. said).

Aside: I don't like using the distance function in this definition. If $S$ happens to be empty or $S=\{z\}$, we are led to ponder the meaning of the distance between a point and an empty set. Is the distance undefined in this case? Is it $+\infty$? Is it $+\infty$ even when $X$ is a bounded space? I'd rather not waste my time on such things.

The more common version, "every neighborhood of $z$ contains a point from $S\setminus \{z\}$" does not have this problem, and immediately generalizes to topological spaces.

Definition of accumulation points

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