# Limit point query

#### Boromir

##### Banned
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

##### MHB Seeker
Staff member
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

##### Banned
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

##### MHB Seeker
Staff member
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

##### Well-known member
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.