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Is every point of an open set E contained in R2 a limit point of E?
Wiki User
∙ 11y ago
In reply to "limit point", posted by Jennifer on Sept 24, 2004:
>I have a basic question. Thanks a lot.
>
>Is every point of every open set E (is contained in R^2) a limit point of E?
Yes, it is. If O is open and x is in O then some open ball B(x,e) is contained in O
(as x is an interior point of O).
But open balls in R^2 have the property that they contain many more points than just x
(eg also (x_1 + 1/2*e, x_2) for x = (x_1, x_2), and e>0 ) and so if B(x,r) is any neighbourhood
of x, then B(x, min(r,e)) will contain this point, which is in O (as B(x,e) \subset O) and not equal to x.
So x is a limit point of O.
>In case of for clsed sets in R^2?
>
There it fails, eg if C = {(1/n, 0): n in N}.
No point of C is a limit point of C (but (0,0) is), as is easily checked.
Henno
Wiki User
∙ 11y ago
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