closure or adherence

O

oumaima1

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Sep 1, 2014
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Hello i'm stuck here
I'm given this definition: Consider (E,S) a topological space and A a part of E.
We name a closure or adherence of A the smallest closed set containing A.
The question:
Consider a part of E and X∊E,and A' the set of the accumlation points of A.
Show that C = A' U A (C is the closure of A)
 
oumaima1 said:
Hello i'm stuck here
I'm given this definition: Consider (E,S) a topological space and A a part of E.
We name a closure or adherence of A the smallest closed set containing A.
The question:
Consider a part of E and X∊E,and A' the set of the accumlation points of A.
Show that C = A' U A (C is the closure of A)
Click to expand...
What have you tried? How far have you gotten? Where are you stuck?

For instance, set equality is often proven through double-set-inclusion, being the proof that the one side is a subset of the other, and the other is a subset of the one. This subset-hood is often proven by "chasing elements"; that is, picking a generic element of the one side and showing that it must also be an element of the other side.

How far have you gotten in this process?

Please be complete. Thank you! ;)
 
oumaima1 said:
Hello i'm stuck here
I'm given this definition: Consider (E,S) a topological space and A a part of E.
We name a closure or adherence of A the smallest closed set containing A.
The question: Consider a part of E and X∊E,and A' the set of the accumulation points of A.
Show that C = A' U A (C is the closure of A)
Click to expand...
There is some disagreement on the definition of accumulation point.
Can you show that \(\displaystyle C\) is a closed set?
Does it contain \(\displaystyle A\)?
Can a smaller closed set contain \(\displaystyle A~?\)
 
stapel said:
What have you tried? How far have you gotten? Where are you stuck?

For instance, set equality is often proven through double-set-inclusion, being the proof that the one side is a subset of the other, and the other is a subset of the one. This subset-hood is often proven by "chasing elements"; that is, picking a generic element of the one side and showing that it must also be an element of the other side.

How far have you gotten in this process?

Please be complete. Thank you! ;)
Click to expand...
I have already tried what you told me but didn't work,it's too difficult !
 
Last edited:
stapel said:
...set equality is often proven through double-set-inclusion.... This subset-hood is often proven by "chasing elements".... How far have you gotten in this process?
Click to expand...
oumaima1 said:
I have already tried what you told me but didn't work,it's too difficult !
Click to expand...
What did you try? How far did you get? At what point did you bog down?

Please be complete when showing your work either for this method or else in reply to the questions other helpers have asked of you. Thank you! ;)
 
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