- #1
Damidami
- 94
- 0
I'm working on some topology in [itex] \mathbb{R}^n [/itex] problem, and I run across this:
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!
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!