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This question already has an answer here:

Prove $B(x,\delta)$ is open.

What this question is asking me to prove...? I don't understand nor have a clue to approach the question...

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marked as duplicate by Stefan Hamcke, Michael Albanese, Ayman Hourieh, Cameron Buie, Pedro Tamaroff Aug 22 '13 at 0:55

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Hint: what is the definition of an open set? $\endgroup$ – Seub Aug 22 '13 at 0:36
  • $\begingroup$ Remember that $B(x,\delta)$ is a set itself. You want to show that something is a subset of another set. $\endgroup$ – Hawk Aug 22 '13 at 0:39
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    $\begingroup$ Here you asked about the Hausdorff dimension.. I am confused. $\endgroup$ – Pedro Tamaroff Aug 22 '13 at 0:45
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    $\begingroup$ Don't let your math.SE account to your younger sister... $\endgroup$ – Gaston Burrull Aug 22 '13 at 1:13
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An open set $U$ is one such that for each point $p\in U$ there is a ball $B(p,\epsilon)\subset U$.

You're asked to prove that $B(x,\delta)$ is an open set using the definition above.

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To prove that $B(x,\delta)$ you should show that for every $y\in B(x,\delta)$ we find $r>0$ such that $$B(y,r)\subset B(x,\delta)$$

Hint Draw a picture to see how to find $r$.

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Let $p\in B(x,\delta)$ so $d(p,x)<\delta$ in which $d$ stands for our meter. If we set $$\epsilon=\delta-d(x,p)$$ which is positive so you can easily show that the open ball, say $B'(p,\epsilon)$ is a subset of $B(x,\delta)$. This satisfies the definition of a open set.

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