I am reading "Measure, Integration & Real Analysis" by Sheldon Axler.
The author wrote the following sentence.
Sentence 1:
Now we can use the Cantor function to show that the Cantor set is uncountable even though it is a closed set with outer meaure $0.$
Why did the author write "closed" in the above sentence?
The following sentence is also true.
But the author didn't write the following sentence.
Sentence 2:
Now we can use the Cantor function to show that the Cantor set is uncountable even though it is a set with outer meaure $0.$
Would you be less surprised if the author wrote Sentence 2 instead of Sentence 1?