Prove or disprove that closure of union of infinitely many sets is equivalent to union of infinitely many closure.

Question

Let $\overline{A},\overline{B}$ denote the closure of a set $A,B\subseteq \mathbb{R}$ respectively.

Prove or disprove that \begin{align*} \bigcup_{n=1}^{\infty}\overline{A_{n}} = \overline{\bigcup_{n=1}^{\infty} A_n}\end{align*}

Firstly, i can prove that $\overline{A\cup B}=\overline{A}\cup \overline{B}$.

I then can use this fact to prove that the main statement is true by Mathematical Induction.

However, i can also find a counter-example by letting $A_{n}=[\frac{1}{n},1].$

So, i think the method by Mathematical Induction must be wrong. But i could not see how so.

See the remark given by Cameron in here.You will see there that the above equality does not hold. — Juniven Acapulco, Mar 03 '17 at 09:47

uniquesolution · Answer 1 · 2017-03-03T09:47:00.410

3

You can use mathematical induction to prove the fact that $\overline{A\cup B}=\overline{A}\cup\overline{B}$ to prove that for every $n$, $$\bigcup_{k=1}^n\overline{A_k}=\overline{\bigcup_{k=1}^nA_k}$$

but you cannot use mathematical induction to prove the corresponding statement for the infinite union.

Your counter-example is correct.

edited Mar 03 '17 at 09:47

answered Mar 03 '17 at 09:42

uniquesolution

18,541

why cant i use mathematical induction for the infinite union? – Little Rookie Mar 03 '17 at 09:44
Please state the principle of mathematical induction. – uniquesolution Mar 03 '17 at 09:45
i learned mathematical induction can be used to prove a statement for $n\geq k$ for some integer $k$. – Little Rookie Mar 03 '17 at 09:51
That is correct. What is the $n$'th statement you want to prove by induction in case of the infinite union?? – uniquesolution Mar 03 '17 at 09:55
i would prove the statement is true for $n=k$, then assume it is true for $n=m$ for some $m\geq k ,\in \mathbb{Z}$, show it is also true for $n=m+1$. – Little Rookie Mar 03 '17 at 09:58
Yes, but my point is that the infinite union is not a statement about any particular integer $n$, and therefore it is not covered by mathematical induction. Do you see it now? – uniquesolution Mar 03 '17 at 09:59
1

Are u saying that Mathematical Induction can only be applied to $n\in \mathbb{N}$ and not uncountable infinite set?
I don't understand what you are trying to express.
– Little Rookie Mar 03 '17 at 10:03

score 3 · Answer 2 · answered Mar 03 '17 at 09:45

3

Induction is a little trickier than that, I actually made the same mistake when I first learned it.

Induction says that a statement is true for all $n\in \Bbb N$, but is says nothing about an infinite number. For example $A\cap B$ is open for open $A$ and $B$, so by induction any finite intersection of open sets is open. Same thing with your theorem, you have shown any closure of a finite union is the union of closures.

answered Mar 03 '17 at 09:45

Good point. ${}{}$ – Giuseppe Negro Mar 03 '17 at 09:48
Isnt $\mathbb{N}$ an infinite set? – Little Rookie Mar 03 '17 at 10:09
@LittleRookie yes but it does not have an infinite element – Mar 03 '17 at 19:25
@Noah how can an element be infinite? – Little Rookie Mar 04 '17 at 04:19
@LittleRookie there is something called transfinite induction on the ordinal numbers, which has an element $\omega$ which is greater than all natural numbers. All induction on the natural numbers says is that for any element in the naturals, the statement is true for that element. Thus to prove an infinite union has a given property, there would need to be an infinite element for the statement to be true at. – Mar 04 '17 at 09:02
Why can't i prove that it is true for all natural numbers? After all, $\mathbb{N}$ contains infinite many elements. – Little Rookie Mar 04 '17 at 09:05
It is true for all natural numbers. $\infty$ is not a natural number. – Mar 04 '17 at 17:00
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"$n$ is finite" is true for all $n \in \mathbb N$. Yet, fails for $\infty$. Induction only works when there are finitely many intermediary steps between cases. Here is a potentially helpful link – Andres Mejia Mar 07 '17 at 18:36

Prove or disprove that closure of union of infinitely many sets is equivalent to union of infinitely many closure.

2 Answers2