# Union and Intersection of Neighborhoods

• Dec 1st 2009, 06:16 AM
dubito
Union and Intersection of Neighborhoods
Let $\epsilon>0, \delta>0$ and $a \in \mathbb{R}$. Show that $V_\epsilon(a) \cap V_\delta(a)$ and $V_\epsilon(a) \cup V_\delta(a)$ are $\gamma$-neighborhoods of $a$ for appropriate values of $\gamma$.

Any hints will be appretiate. I have a proof, but it's really simple and I doubt it's correct. Thanks!
• Dec 1st 2009, 02:46 PM
Drexel28
Quote:

Originally Posted by dubito
Let $\epsilon>0, \delta>0$ and $a \in \mathbb{R}$. Show that $V_\epsilon(a) \cap V_\delta(a)$ and $V_\epsilon(a) \cup V_\delta(a)$ are $\gamma$-neighborhoods of $a$ for appropriate values of $\gamma$.

Any hints will be appretiate. I have a proof, but it's really simple and I doubt it's correct. Thanks!

What is a $\gamma$ neighborhood.
• Dec 19th 2009, 07:49 AM
dubito
Don't know (Wondering) that's all the information I was given.
• Dec 19th 2009, 08:30 AM
Plato
Quote:

Originally Posted by dubito
Let $\epsilon>0, \delta>0$ and $a \in \mathbb{R}$. Show that $V_\epsilon(a) \cap V_\delta(a)$ and $V_\epsilon(a) \cup V_\delta(a)$ are $\gamma$-neighborhoods of $a$ for appropriate values of $\gamma$.

Let $\gamma = \min \left\{ {\delta ,\varepsilon } \right\}\,\& \,\Gamma = \max \left\{ {\delta ,\varepsilon } \right\}$.
Under the usual undestanding of $\gamma-\text{neighborhoods}$,
$V_\epsilon(a) \cap V_\delta(a)=V_{\gamma}(a)$ and $V_\epsilon(a) \cup V_\delta(a)=V_{\Gamma}(a)$.
• Dec 20th 2009, 08:17 AM
dubito
Thanks that's similar to what I had.