I am studying Lemma 10.2 in Munkres' Topology and I think I understand the proof but there is something I clearly don't understand about minimal uncountable well-ordered set:
Let A be a minimal uncountable well-ordered set. Here $A$ = $S_\Omega$ $\cup$ $\Omega$. From my understanding of the Lemma and the paragraph that follows it, $S_\Omega$ is uncountable but $S_t$ (where $t<\Omega$) is countable. What makes having this element $\Omega$ make the section of $A$ by $\Omega$ uncountable where as every other section is countable? It seems like, in naive tongue, $\Omega$ destroys the "countability" of A somehow - had I defined my $A$ without this $\Omega$, then every section of $A$ would have been countable.
(1) Hence, my question - is there an example I can look at and understand this?
Addendum:
(2) Does it mean that an uncountable set can have subsets that are countable?
A set of a certain cardinality has subsets of all smaller cardinality, so in particular an uncountable set has countable subsets (many of them in fact).
If we give an uncountable subset like $\Bbb R$ a well-order (which we can do, using AC, the Axiom of Choice/Zorn's Lemma or an equivalent formulation), we get an order $<_w$ on $\Bbb R$ (totally different from the usual order on $\Bbb R$, probably) and we can consider the size of sections of $<w$: Recall that the section of $x$ is defined as $S_x= \{y\in \Bbb R\mid y <_w x\}$; $S_x$ is empty if $x$ is the minimum $m$ of $<_w$ (which exists) and size $1$ if $x$ is the successor of that minumum (i.e. the minimum of all $x$ that are $>m$, which also exists etc.):
Either all sections of all $x$ are countable and we can use $(\Bbb R,<_w)$ as $\Omega$ (in Munkres' notation) or $\{x\mid S_x \text{ is uncountable }\}$ is a non-empty subset of $\Bbb R$ and so (!) has a minimal element which we can call $\Omega$. So $\Omega$ has a uncountable section, but all $x <_w \Omega$ have a countable section, or else this would contradict the minimality of $\Omega$.
Note that this is not at all a concrete example as no easy description of a well-order on $\Bbb R$ can exist (for all sorts of reasons related to models of set theory without AC). But maybe it heps to think about it bit more: $\Omega$ (or $\omega_1$, as is the more common denotation) "lives inside" every uncountable well-ordered set (as hard as these are to visualise; but believe in AC and they exist!)..