accumulation points

[logistic_guy]

Prove that if a set contains each of its accumulation points, then it must be a closed set.

 

[khansaheb]

Prove that if a set contains each of its accumulation points, then it must be a closed set.

Please define "accumulation points" using a numerical example.

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

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Please share your work/thoughts about this problem

 

[logistic_guy]

This problem is a little bit difficult, so I will answer it in steps. If one of my steps doesn't make sense, stop me and I will try to correct it.

For now let us assume that we have a set $\displaystyle \text{A}$ which contains complex elements.

 

[khansaheb]

Please define "accumulation points" using a numerical example - as it is presented in your text book.

 

[logistic_guy]

Please define "accumulation points" using a numerical example - as it is presented in your text book.

Thank you Sir khan.

 

[logistic_guy]

This problem is a little bit difficult, so I will answer it in steps. If one of my steps doesn't make sense, stop me and I will try to correct it.

For now let us assume that we have a set $\displaystyle \text{A}$ which contains complex elements.

I will assume that set $\displaystyle \text{A}$ contains all of its accumulation points.

 

[pka]

Prove that if a set contains each of its accumulation points, then it must be a closed set.

This is an interesting and often used problem. As to the definition that depends upon the textbook in use.
Two major texts are: 1) by Hocking & Young (it in the R.L. Moore school) & 2) the other by R L Kelly.
The first does not even mention accumulation points. Kelly on the other hand
defines an accumulation point, $\alpha$, of a set $A$ to be the same as a limit point or cluster point.
So each open set $O$ that contains $\alpha$ must also contain a point of $A$ distinct from $\alpha$.

 

[logistic_guy]

This is an interesting and often used problem. As to the definition that depends upon the textbook in use.
Two major texts are: 1) by Hocking & Young (it in the R.L. Moore school) & 2) the other by R L Kelly.
The first does not even mention accumulation points. Kelly on the other hand
defines an accumulation point, $\alpha$, of a set $A$ to be the same as a limit point or cluster point.
So each open set $O$ that contains $\alpha$ must also contain a point of $A$ distinct from $\alpha$.

Thank you pka. But the accumulation points in the op context are related to Complex Analysis! Maybe it doesn't matter.

Look for example at these two sets:

$\displaystyle A = \{ z \in \mathbb{C} : |z| \leq 1 \}$

$\displaystyle B = \{ z \in \mathbb{C} : |z| < 1 \}$

I know that set $\displaystyle A$ contains all of its accumulation points while set $\displaystyle B$ doesn't. The idea is simple but how to prove it is very difficult!