Recently, I have read a Schaum's book. General Topology, and it introduced the concept of cardinality and order, but it points out something that I really don't understand.
If $A\preceq B$ and $B\preceq A$, then $A \sim B$.
If $X\supseteq Y\supseteq X_1$ and $X\sim X_1$, then $X\sim Y$
Prove these statements are equivalent which are both Schroeder-Berstein Theorem. In fact, I know how to prove them separately but is there a easy way to prove equivalent?
2.In the book, it mentions 'axiom of choice' a lot. So I search it from wiki and I found that it means: $$\forall X[\emptyset\notin X\Rightarrow \exists f:X\rightarrow\cup X \forall A\in X(f(A)\in A)]$$
It is quite easy to understand. But the book said this is equivalent to Zorn's Lemma:
Let $X$ be a non-empty partially ordered set in which every totally ordered subset has an upper bound. Then X contains at least one maximal element.
Wiki also mention that $\aleph_0$ is the smallest cardinality of a infinite set. This is directly derive from axiom of choice. Both of them I don't understand why.
3.Prove Law of trichotomy: Given any pair of set , either $A\prec B$ , $B\prec A$, or $A\sim B$. It said it can be proved ny using Zorn's Lemma.