- #1
Mogarrr
- 120
- 6
Homework Statement
Let [itex] A_1, A_2, A_2,...[/itex] be subsets of a metric space.
(a) If [itex]B_{n}=\bigcup_{i=1}^{n}A_{i} [/itex], prove that [itex] \bar{B_{n}}= \bigcup_{i=1}^{n}\bar{A_{i}} [/itex] for n=1,2,3,...
(b) If [itex]B=\bigcup_{i=1}^{\infty}A_{i} [/itex], prove that [itex] \bar{B} \subset \bigcup_{i=1}^{\infty}\bar{A_{i}} [/itex],
Show by example this inclusion is proper.[/B]
Homework Equations
If E is a subset of a metric space, then the closure of E,
[itex] \bar{E}=E \bigcup \acute{E} [/itex], where [itex] \acute{E} [/itex] is the set of all limit points of E.
For a subset E of a metric space X, a point [itex] p \in X [/itex] is a limit point if and only if
[itex] \forall \epsilon > 0, \exists q \in B(p;\epsilon) \bigcap E : q \neq p [/itex]
The Attempt at a Solution
Now I think I've got part (a) and (b) nailed down, but I'm having trouble thinking of an example for a proper subset. Here's what I have thus far...
Suppose [itex]x \in \bar{B_{n}} [/itex], then [itex] x \in B_{n} [/itex] or x is a limit point for [itex] B_{n} [/itex].
If [itex]x \in {B_{n}} [/itex], then [itex]x \in \bigcup_{i=1}^{n} A_{i} [/itex], then [itex] \exists i, x \in A_{i} [/itex], so [itex] \exists i, x \in \bar{A_{i}} [/itex].
If x is a limit point, then [itex] \forall \epsilon > 0, \exists y \in B(x;\epsilon) \bigcap B_{n}: y \neq x [/itex], so [itex] \forall \epsilon > 0, \exists y \in B(x;\epsilon) \bigcap (\bigcup_{i=1}^{n} A_{i}): y \neq x [/itex], then
[itex] \exists i,\forall \epsilon > 0, \exists y \in B(x;\epsilon) \bigcap A_{i}: y \neq x [/itex], then
[itex] \exists i, x\in \bar{A_{i}}[/itex], so [itex] x \in \bigcup_{i=1}^{n}\bar{A_{i}}[/itex]
Suppose [itex] x \in \bigcup_{i=1}^{n}\bar{A_{i}}[/itex], then [itex] \exists i, x \in A_{j}[/itex] or for some i x is a limit point.
If [itex] \exists i, x \in A_{i}[/itex], then [itex] x \in \bigcup_{i=1}^{n}A_{i}=B_{n}[/itex].
If x is a limit point for some A sub i, then x is a limit point for [itex] \bigcup A_{i}=B_{n} [/itex].
For part (b), I did the same thing, but replaced [itex] B_{n} [/itex] with [itex] B [/itex] and in the union I replaced n with infinity.
Now, I'm having some difficulty coming up with an example that would show B is a proper subset of the union of an infinite number of A sub i's.
Any hints or corrections would be much appreciated.