Is every point of every closed set E subset of R^2 a limit point of E?

rumjum

1. The problem statement, all variables and given/known data

If E is subset of R^2, then is every point of every closed set E, a limit point of E?

2. Relevant equations



3. 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.

HallsofIvy

One example does not a proof make!

But one counter-example does. Consider the set
{[itex](x,y)| x^2+ y^2\le r[/itex]}[itex]\cup[/itex] {[itex](0, r+1)[/itex]}

Is (0, r+1) a limit point of that set?

Look up "isolated point" in your text book.

rumjum

What does that mean?

(0,r+1) lies outside the set E. So, we can find points in the neighborhood of (0,r+1) such that the intersection with E is null. If I understand you correctly, you are saying that since the points outside E are not limit points and E is a closed set, so points of E need to be limit points. Or E should not have any limit point to be a closed set. And so on.....