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Rudin's Theorem 2.27 
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#1
Jan213, 09:42 PM

P: 376

Prove the closure of E in a Metric Space X is closed. (page 35)
Rudin states: if p∈X and p∉E then p is neither a point of E nor a limit point of E.. Hence, p has a neighborhood which does not intersect E. (Great) The compliment of the closure of E is therefore open. WHY? I don't see it... BTW, I know there are different ways to proving this, but I want to understand the last line jump. Thanks. 


#2
Jan213, 09:46 PM

P: 772




#4
Jan213, 09:49 PM

P: 376

Rudin's Theorem 2.27
The problem we should say that the complement of E is open, not the complement of the closure of E. 


#5
Jan213, 10:08 PM

P: 376

I guess since the intersection of N(p) and E is empty then no point q of N(p) can be a limit point of E as this would mean every neighborhood of q will contain an infinite number of points in E. Hence the intersection of N(p) and "closure of E" is empty.
Is this correct? 


#6
Jan213, 11:31 PM

Sci Advisor
P: 1,716

If any point not in E has an open neighborhood that does not intersect E then by definition the complement of E is open.



#7
Jan213, 11:48 PM

P: 376

my bad, I forgot to add the word: "closure" in the last line of the proof. I just reread it.
This is what is confusing me: FROM: Hence, p has a neighborhood which does not intersect E. We get: The compliment of the closure of E is therefore open. 


#8
Jan313, 06:34 AM

P: 261

We have just shown that p is in the complement of the closure of E (call it A). We also showed that p has a neighborhood that is entirely in A. Hence, p is an interior point of that set A. Hence A is open.



#9
Jan313, 09:59 AM

P: 345

Then, everthing is clear. 


#10
Jan313, 03:28 PM

P: 376

Let ##N(p)## be the neighborhood with no common points with ##E##. What about ##\overline E##? Is the ##\overline E \cap N(p)## an empty set because if it wasn't, then ##N(p)## will contain a limit point of ##E## and these will have neighborhoods that contain a point of ##E##? "I understand everything about the proof, except for the part where we go from ##E## to ## \overline E## when we mention the complement. I want to make sure my reasoning is correct" Thanks. 


#11
Jan313, 04:11 PM

P: 345




#12
Jan313, 05:06 PM

P: 376




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