# Can open balls and pairwise intersections of balls generate a topology?

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• Oct 7th 2011, 02:06 PM
algorytmus
Can open balls and pairwise intersections of balls generate a topology?
Hi,

I have a question. Can the following topology base generate a topology(a family of sets)?:

X - finite set of points, d - metric
B - open ball in X
r - ray, const value

Additionally, how could be the cover of a family of sets in topology realized ?

Thans in advance,
algorytmus
• Oct 7th 2011, 02:41 PM
hatsoff
Re: Can open balls and pairwise intersections of balls generate a topology?
A family $\displaystyle \mathcal{F}$ of sets generates a topology on $\displaystyle X$ just in case:

(1) $\displaystyle \mathcal{F}\subseteq 2^X$;
(2) $\displaystyle x\in X$ $\displaystyle \implies$ $\displaystyle \exists B\in\mathcal{F}$ such that $\displaystyle x\in B$;
(3) $\displaystyle x\in B_1\cap B_2$ for $\displaystyle B_1,B_2\in\mathcal{F}$ $\displaystyle \implies$ $\displaystyle \exists B_3\in\mathcal{F}$ such that $\displaystyle x\in B_3\subseteq B_1\cap B_2$.

Your collection satisfies these requirements. In fact, notice that the second term of the union is superfluous, and that $\displaystyle \mathcal{B}=\{B_d(x,r):x\in X,r>0\}$. Given that $\displaystyle X$ is finite, this means $\displaystyle \mathcal{B}=2^X\setminus\emptyset$, i.e. it generates the discrete topology.

I'm not sure what you mean by realizing a cover of a family of sets though.
• Oct 7th 2011, 03:21 PM
algorytmus
Re: Can open balls and pairwise intersections of balls generate a topology?
Thank you for help!

Could you correct me if I am wrong in the following:

Let me illustrate examples geometrically.

Given X={a, b, c}
https://lh3.googleusercontent.com/-t...Jk/s262/X1.png

the topological base is

$\displaystyle \mathcal{B}$={{B_a},{B_b},{B_c},{B_a,B_b},{B_b,B_c}}

,
for
https://lh4.googleusercontent.com/-J...jg/s286/X2.png

it is:

$\displaystyle \mathcal{B}$={{B_a},{B_b},{B_c}}

B_x - a ball for x
• Oct 7th 2011, 04:58 PM
hatsoff
Re: Can open balls and pairwise intersections of balls generate a topology?
Well if you mean that $\displaystyle \mathcal{B}=\{\{x\}:x\in X\}$, then it still generates the discrete topology.