Let(X, d) be a metric space. the set {y ∈ X : d(x, y) ≤ r} is a closed ball centered at X and with radius r.

(a)Show that a closed ball is a closed set.

Printable View

- Nov 19th 2009, 03:19 PMhebbyMetric Space, closed sets in a closed ball
Let(X, d) be a metric space. the set {y ∈ X : d(x, y) ≤ r} is a closed ball centered at X and with radius r.

(a)Show that a closed ball is a closed set. - Nov 19th 2009, 04:43 PMFocus
Have you tried any of the questions you posted? If so post your working out. A tip to get you started, pick a point outside of the ball and find and open ball that contains it.

- Nov 19th 2009, 05:30 PMDrexel28
Denote the closed ball centered at $\displaystyle x$ of radius $\displaystyle r$ as $\displaystyle B_r(X,x)$. Now suppose that $\displaystyle \xi$ was a limit point of $\displaystyle B_r(X,x)$ but not an element of $\displaystyle B_r(X,x)$. Then every open ball around $\displaystyle \xi$ would contain another point of $\displaystyle B_r(X,x)$ besides $\displaystyle \xi$. Therefore $\displaystyle d(x,\xi)\le r+\varepsilon\quad\forall\varepsilon>0$. Assume that $\displaystyle d(x,\xi)>r\implies d(x,\xi)-r>0$, then choosing $\displaystyle \varepsilon=d(x,\xi)-r$ would derive a contradiction. Therefore $\displaystyle d(x,\xi)\le r$ and the conclusion follows.