query: What do I have to suppose to say an infinite set has a proper infinite subset and its complement is also infinite?
positive_passages.docid: 19414
positive_passages.text: Uncountable subset with uncountable complement, without the Axiom of Choice
negative_passages.docid: 119624
negative_passages.text: A Dedekind infinite set has a countably infinite proper subset
negative_passages.docid: 82009
negative_passages.text: Prove that every totally ordered set has a well-ordered cofinal subset
negative_passages.docid: 50258
negative_passages.text: How can a set contain itself?
negative_passages.docid: 86926
negative_passages.text: Why isn't this a well ordering of $\{A\subseteq\mathbb N\mid A\text{ is infinite}\}$?
negative_passages.docid: 132537
negative_passages.text: Class of sets of a given infinite cardinality
negative_passages.docid: 9651
negative_passages.text: Cardinality of cartesian product of an infinite set with N
negative_passages.docid: 2403
negative_passages.text: Can a countable set contain uncountably many infinite subsets such that the intersection of any two such distinct subsets is finite?
negative_passages.docid: 24372
negative_passages.text: Infinite Set is Disjoint Union of Two Infinite Sets
negative_passages.docid: 120712
negative_passages.text: Is there a way to define the "size" of an infinite set that takes into account "intuitive" differences between sets?
negative_passages.docid: 20706
negative_passages.text: Non-existence of a Surjective Function from a Set to Its Subsets (Cantor's theorem)
negative_passages.docid: 97064
negative_passages.text: Dense set in an infinite cofinite space
negative_passages.docid: 77548
negative_passages.text: Given an arbitrary infinite set $S$, how to construct an injection $\mathbb{N} \to S$ with the Axiom of Choice?
negative_passages.docid: 66398
negative_passages.text: Union of a finite set and a countably infinite set is countably infinite
negative_passages.docid: 60678
negative_passages.text: Does 'A countable infinite Cartesian product of countable sets is not necessarily countable' violate induction principle?
negative_passages.docid: 146560
negative_passages.text: Is there any uncountably infinite set that does not generate the reals?
negative_passages.docid: 91163
negative_passages.text: Let the set S be infinite, and the set T countably infinite. Show that S and S U T have the same cardinality