Contains closed set Accumulation points?

1. Homework Statement

Hello, I am here a novice and my English is very bad. I am a student and now we learning about sets. I have got a problem, how to proof, that every closed set contains all accumulation points? I know / hope, that should, but I want to proof it. I hope, that somebody will help me. Have a nice day, Michal, Slovakia

2. Homework Equations

3. The Attempt at a Solution

Answers and Replies

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HallsofIvy
Science Advisor
Homework Helper
What definition of "closed set" are you using? There are several different but equivalent definitions of "closed set"- and in mathematics, proofs often use the specific words of definitions. No one can help you prove this without knowing what definition you are using.

What definition of "closed set" are you using? There are several different but equivalent definitions of "closed set"- and in mathematics, proofs often use the specific words of definitions. No one can help you prove this without knowing what definition you are using.
I do not know, how do you think it, but this is example for my closed set

Closed interval [a,b] is closed subset of real numbers

We have not in school definition of closed and opened set yet. But we are working with functions too, also it should be a definition of intervals of the function -set of function of definite domain? I really do not know. I hope, that do you understand me. Thanks for your fast answer

HallsofIvy
Science Advisor
Homework Helper
If you do not know the definition of "closed set", then you cannot possibly prove anything about them! Examples will not provide proofs. Since you have a specific definition of "closed interval" you might be able to prove that a closed interval contains all its accumulation points. What is your definition of "accumulation point"?

If you do not know the definition of "closed set", then you cannot possibly prove anything about them! Examples will not provide proofs. Since you have a specific definition of "closed interval" you might be able to prove that a closed interval contains all its accumulation points. What is your definition of "accumulation point"?
Let (zero) ∅≠M⊂R. Point a∈R is accumulation point of set M, if for every O_{ε}°(a) exists x∈M, x∈O_{ε}°(a).

I hope that there will be no problem with syntax, because I am not using LaTex yet