I'm working on some topology in [itex] \mathbb{R}^n [/itex] problem, and I run across this:(adsbygoogle = window.adsbygoogle || []).push({});

Given [itex]\{F_n\}[/itex] a family of subsets of [itex] \mathbb{R}^n [/itex], then if [itex]x[/itex] is a point in the clausure of the union of the family, then

[itex]x \in \overline{\cup F_n}[/itex]

wich means that for every [itex]\delta > 0[/itex] one has

[itex]B(x,\delta) \cap (\cup F_n) \neq \emptyset [/itex]

Now, if I say that because the intersection is a nonempty set, I have a point [itex]y \in \mathbb{R}^k [/itex] such that

[itex] d(x,y) < \delta \wedge y \in (\cup F_n)[/itex]

did I use the axiom of choice?

Because I think it this way, I choose an element [itex] y [/itex] from an infinite (nonumerable) indexed family parametrized by [itex]\delta[/itex] (for each value of [itex]\delta[/itex] I have another intersection).

So If I don't want to use the AC I can't just say that I can choose an element from that nonempty intersection?

I'm a little confused :?

Thanks!

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# Sets intersection and the axiom of choice?

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