countable set and disjoint subsets

Asked Oct 21 '21 at 08:59

Active Oct 21 '21 at 09:53

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Let $A_1,A_2,...,A_n$ disjoint subsets of an infinite set $\Omega$. If $\Omega$ is countable then there can only be at most one infinite disjoint subset in $A_1,A_2,...,A_n$. However if $\Omega$ is uncountable then there can be more than one infinite disjoint subset in $A_1,A_2,...,A_n$.

Is this statement always true?

If true, this is it also true for $A_1,A_2,...$ infinite number of disjoint subsets?

edited Oct 21 '21 at 09:53

Asaf Karagila

393,674

asked Oct 21 '21 at 08:59

user3911153

1

Welcome to MSE. You should choose your tags carefully. What has this to do with [tag:functional-analysis]? – José Carlos Santos Oct 21 '21 at 09:01
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$\Omega={1,2,3,4,...}, A_1={1,3,5...}, A_2={2,4,6,...}$. – Kavi Rama Murthy Oct 21 '21 at 09:02
@KaviRamaMurthy, Is there countable infinite set with $n$ disjoint sets are infinite? – user3911153 Oct 21 '21 at 09:06
@JoséCarlosSantos, this is the subject we are studying now. – user3911153 Oct 21 '21 at 09:06
But your question has nothing to do with Functional Analysis. – José Carlos Santos Oct 21 '21 at 09:07
Lecture notes say either all disjoint subsets are finite or at least one is infinite but not all infinite. – user3911153 Oct 21 '21 at 09:08
@JoséCarlosSantos, tag removed. – user3911153 Oct 21 '21 at 09:09
Consider ${p,p^{2},p^{3},...}$ where $p$ ranges over all prime numbers. – Kavi Rama Murthy Oct 21 '21 at 09:11

countable set and disjoint subsets

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