about limit point


by jwqwerty
Tags: limit, point
jwqwerty
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#1
Mar23-12, 03:06 PM
P: 23
the definition of limit point:
a point p is a limit point of E(subset of metric space X) if every neighborhood of p contains a point q≠p which is in E.

My question is that is there a limit point p which is not in E?
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SteveL27
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#2
Mar23-12, 03:10 PM
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Quote Quote by jwqwerty View Post
the definition of limit point:
a point p is a limit point of E(subset of metric space X) if every neighborhood of p contains a point q≠p which is in E.

My question is that is there a limit point p which is not in E?
Take the open unit interval E = (0,1) as a subset of the real numbers. Can you think of a real number that's not in (0,1) but that satisfies the definition of limit point of E? (Hint: Can you think of TWO such points?)
jwqwerty
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#3
Mar23-12, 03:31 PM
P: 23
Quote Quote by SteveL27 View Post
Take the open unit interval E = (0,1) as a subset of the real numbers. Can you think of a real number that's not in (0,1) but that satisfies the definition of limit point of E? (Hint: Can you think of TWO such points?)
thanks but i have another question

can you give me an example of a set that is perfect?
def: E is perfect if E is closed and if every point of E is a limit point of E

HallsofIvy
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#4
Mar23-12, 03:59 PM
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about limit point


So you refuse to answer SteveL27's question?

But I will answer your question: [0, 1].

It's actually harder to give an example of a closed set that is NOT perfect. Can you?
Alesak
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#5
Mar25-12, 02:11 PM
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Quote Quote by HallsofIvy View Post
It's actually harder to give an example of a closed set that is NOT perfect. Can you?
What stops me from adding {0} to usual topology of real line, so that isīs open set? Then (0, 1] would be closed and not perfect. Certainly not easy, I canīt think of any more standard example.
alexfloo
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#6
Mar25-12, 07:24 PM
P: 192
(0, 1] is not closed; it's just also not open. A good example of a closed non-perfect set is one with an isolated point, like {2}, or [0,1][itex]\cup[/itex]{2}.


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