- #1
rumjum
- 40
- 0
Homework Statement
If E is subset of R^2, then is every point of every closed set E, a limit point of E?
Homework Equations
The Attempt at a Solution
I think the answer is yes. Consider E = { (x,y) | x^2 + y^2 <= r^2} , where r is the radius.
Consider a point p that belongs to E, then p shall be a limit point if
the intersection of Ne(p) ( that is neiborhood of "p" with "e" as radius) and set E has another point "q", such that p and q are not the same.
Now, we know that the Ne(p) = circle with radius "e" around "p". Since "p" is an internal point the intersection of this circle with that of E, (another circle) shall have several points other than "p". Hence, all points in E are limit points.
any comments? Thanks.