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Limit point query

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Boromir

Banned
Feb 15, 2014
38
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
Mar 5, 2012
8,774
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.
 
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Boromir

Banned
Feb 15, 2014
38
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
Mar 5, 2012
8,774
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
Jan 26, 2012
236
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.