Can the union of uncountable infinite sets be a countable infinite set?

Question

Can the union of uncountable infinite sets be a countable infinite set?

If there is such a set, I would be grateful if you answer the question by giving an example.

If the question is too simple, I apologize.

Answer 1

Let $A$ be one of the uncountable sets, and let $B$ be their union.

Suppose $B$ is countable.

By the definition of set union, every element of $A$ is also an element of $B$: that is, $A\subseteq B$.

Therefore $B$ is a countable set with an uncountable subset, which is a contradiction.

Therefore $B$ cannot be countable.

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Note: This relies on the theorem that every subset of a countable set is countable. The easiest way to see why that's true is to imagine counting the original set but skipping elements which aren't in the subset. A proof just involves expressing that more formally.

Answer 2

Unions cannot decrease the cardinality of a set, so suppose $A_i$ are uncountable then the finite union has cardinality: $$ |\bigcup\limits_{i=0}^{N} A_i|\geq |A_i|$$ for any of the $A_i$. Which means they are definitely uncountable.

Answer 3

Maybe the OP meant the intersection of uncountable sets. If so, then certainly it's possible with a countable infinity of input sets. The $p$-adic integers for any prime $p$ are uncountably infinite, but the intersection of all such sets for all primes $p$ is just the ordinary integers.