Limit point query

[Boromir]

If S is a subset of X,a metric space, I have always assumed that the definition of a limit point of S, say x, was that there a sequence in S converging to x. Therefore if x is in S, it is a limit point. That is, every member of S is a limit point of S, just by taking the sequence x,x,x,x,x,..... However on the wiki page, there is no statement that every member of a set is a limit point of the set, which is what I would expect.

Is this a 'correct' definition?

 

[Klaas van Aarsen]

If S is a subset of X,a metric space, I have always assumed that the definition of a limit point of S, say x, was that there a sequence in S converging to x. Therefore if x is in S, it is a limit point. That is, every member of S is a limit point of S, just by taking the sequence x,x,x,x,x,..... However on the wiki page, there is no statement that every member of a set is a limit point of the set, which is what I would expect.

Is this a 'correct' definition?

From wiki:

A point x in X is a limit point of S if every neighbourhood of x contains at least one point of S different from x itself.​

I'm afraid you will need a sequence of points different from x.


Btw, I consider this more Topology than Analysis, so I am moving it there.

 

[Boromir]

From wiki:

A point x in X is a limit point of S if every neighbourhood of x contains at least one point of S different from x itself.​

I'm afraid you will need a sequence of points different from x.


Btw, I consider this more Topology than Analysis, so I am moving it there.

So not every point of a set is limit point? Blast I've been using in a casual way for years. That's completely unintuitive to disbar the constant sequence, what harm can it do?

 

[Klaas van Aarsen]

So not every point of a set is limit point? Blast I've been using in a casual way for years. That's completely unintuitive to disbar the constant sequence, what harm can it do?

Correct.

 

[ThePerfectHacker]

If S is a subset of X,a metric space, I have always assumed that the definition of a limit point of S, say x, was that there a sequence in S converging to x. Therefore if x is in S, it is a limit point. That is, every member of S is a limit point of S, just by taking the sequence x,x,x,x,x,..... However on the wiki page, there is no statement that every member of a set is a limit point of the set, which is what I would expect.

Is this a 'correct' definition?

Here is an example.

Suppose that $X = [0,1]\cup \{2\}$, usual real-line topology.

Every $x$ such that $0\leq x\leq 1$ is a limit point, because we can converge to $x$ along a sequence of points distinct from $x$. However, $2$ is not a limit point, any sequence that converges to $2$ will eventually be a constant sequences of only $2$'s, rather we call $2$ an isolated point.