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shashank dwivedi June 3rd, 2018 10:53 PM

Real Analysis
 
Do the closed set consists of points other than limit points? :confused:
If yes, then what are the examples?

mathman June 4th, 2018 01:33 PM

Isolated points?

shashank dwivedi June 5th, 2018 06:58 AM

Yes points that are apart from limit points in the set. I didn't get what you meant by Isolated points.

cjem June 5th, 2018 07:23 AM

Every point $x$ of a set is a limit point of that set (even if it's an "isolated point"). Indeed, consider the constant sequence $x, x, \dots$.

Maschke June 5th, 2018 07:51 AM

Quote:

Originally Posted by shashank dwivedi (Post 595246)
Yes points that are apart from limit points in the set. I didn't get what you meant by Isolated points.

The set $[0,1] \cup \{2\}$ is closed but $2$ is not a limit point. It's an isolated point. It has a neighborhood that contains no other point of the set besides itself.

cjem June 5th, 2018 11:14 AM

Ah yes, ignore my previous post. I mistakenly took $x$ being a limit point to mean "$x$ is the limit of a sequence of points in the set" (rather than "$x$ is the limit of an eventually non-constant sequence of points in the set"/"every neighbourhood of $x$ contains a point of the set except $x$).

Maschke June 5th, 2018 11:54 AM

Quote:

Originally Posted by cjem (Post 595281)
Ah yes, ignore my previous post. I mistakenly took $x$ being a limit point to mean "$x$ is the limit of a sequence of points in the set" (rather than "$x$ is the limit of an eventually non-constant sequence of points in the set"/"every neighbourhood of $x$ contains a point of the set except $x$).

This is a tricky point (no pun) that confuses everyone.

An adherent point, also known as a point of closure, is a point whose every neighborhood contains some point of the set. So $2$ is a point of closure of $[0,1] \cup \{2\}$. But it's not a limit point, which requires that every neighborhood of the point contains some point of the set other than the point in question.

cjem June 5th, 2018 02:23 PM

Quote:

Originally Posted by Maschke (Post 595283)
This is a tricky point (no pun) that confuses everyone.

An adherent point, also known as a point of closure, is a point whose every neighborhood contains some point of the set. So $2$ is a point of closure of $[0,1] \cup \{2\}$. But it's not a limit point, which requires that every neighborhood of the point contains some point of the set other than the point in question.

Yeah, I'm used to the terms "accumulation point" and "point of closure" for the two concepts. I knew limit point referred to one of the two, but settled on the wrong one without bothering to check!


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