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1. ### jwqwerty

44
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?

2. ### SteveL27

803
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?)

3. ### jwqwerty

44
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

4. ### HallsofIvy

40,421
Staff Emeritus
So you refuse to answer SteveL27's question?

It's actually harder to give an example of a closed set that is NOT perfect. Can you?

5. ### Alesak

134
What stops me from adding {0} to usual topology of real line, so that is´s open set? Then (0, 1] would be closed and not perfect. Certainly not easy, I can´t think of any more standard example.

6. ### alexfloo

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]$\cup${2}.