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Munkres Supplementary Exercises to Chapter 1

* : Supplementary Exercises $1.X$

Theorem
Wosets are Isomorphic to Each Other or Initial Segments

Proof
If the sets $S$ and $T$ considered are empty or singletons, the theorem holds vacuously or trivially.

Thus assume $S$ and $T$ each contains at least two elements.

Let $U = \left({S, \preceq_S}\right) \cup \left({T, \preceq_T}\right)$

Define the following relation $\preceq$ on $U$:


 * $\forall x, y \in U: x \preceq y$




 * $x, y \in S : x \preceq_S y$

or:


 * $x, y \in T: x \preceq_T y$

or:


 * $x \in S, y \in T$

We claim that \preceq is a well-ordering.

First, we show it is a total ordering.

Checking in turn each of the criteria for a total ordering:

Reflexivity
If $x = y$, they're necessarily both in $S$ or $T$ simultaneously.

Reflexivity then follows from $\preceq_S$ and $x \preceq_T$ being reflexive, as they are both orderings.

Transitivity
Let $x, y, z \in U$.

If $x, y, z \in S$ or $x, y , z \in T$ simultaneously, then $\preceq$ is transitive by the transitivity of $\preceq_S$ and $\preceq_T$.

Suppose $x, y \in S$ and $z \in T$.

Let $x \preceq y$ and $y \preceq z$.

Then $x \preceq z$ because $x \in S$ and $y \in T$.

Suppose $x \in S$ and $y, z \in T$.

Then $x \preceq z$ also because $x \in S$ and $y \in T$.

Thus $\preceq$ is transitive.

Antisymmetry
Let $x \preceq y$ and $y \preceq x$.

If $x, y \in S$ then $x = y$ by the antisymmetry of $\preceq_S$.

Likewise if $x, y \in T$.

If $x \in S$ and $y \in T$, then $y \in S$ and $x \in T$ as well.

Thus $x = y$ from the antisymmetry of $\preceq_S$ or $\preceq T$.

Conclude that $\preceq$ is a total ordering.

To show $\preceq$ is a well-ordering, consider a non-empty set $X \subseteq U$.

Then either:


 * $X \cap S = \varnothing$

or:


 * $X \cap T = \varnothing$

or:


 * $X \cap S$ is non-empty and $X \cap T$ is non-empty.

In the first case, $X \subseteq T$, by Intersection with Complement is Empty iff Subset.

Then $X$ has a smallest element defined by $\preceq_T$.

In the second case, $X \subseteq S$, also by Intersection with Complement is Empty iff Subset.

Then $X$ has a smallest element defined by $\preceq_S$.

In the third case, the smallest element of $X \setminus T$ is an element of $S$.

Thus it precedes any element of $T$ by the definition of $\preceq$.

The smallest element of $X \setminus T$, which is a subset of $S$, is guaranteed to exist by the well-ordering on $S$.

This smallest element is then also the smallest element of $\left({X \setminus T}\right) \cup T = X$.

Thus $\preceq$ is a well-ordering on $S \cup T$.

Consider the mapping:


 * $k: \left({T, \preceq_T}\right) \to \left({U, \prec}\right)$:


 * $k(\alpha) = \alpha$

Then $k$ is strictly increasing, by the construction of $\prec$.

Thus there is a strictly increasing mapping from $T$ to $U$.

From Strictly Increasing Mapping Between Wosets Implies Order Isomorphism, $T$ is order isomorphic to $U$ or an initial segment of $U$.

Let $\mathcal I_x$ denote the initial segment in $U$ determined by $x$, according to $k$.

Note that $\mathcal I_x = \mathcal I_{k(x)}$, because $k(x) = x$.

Suppose $x \in S$.

Then $\mathcal I_x \subseteq S$ because $\prec$ is a well-ordering.

Thus there is an order isomorphism from $T$ to $\mathcal I_x$ in $S$.

Suppose $x = \min T$, the smallest element of $T$.

Then every element of $S$ strictly precedes $x$, as $x$ is in $T$.

Also, $x$ precedes every element of $T$, so $\mathcal I_x \ne T$.

Thus there is an order isomorphism from $T$ to all of $S$.

Suppose $x \in T$ and $x \ne \min T$.

Then $\mathcal I_x$ defines an initial segment in $T$.

Also, every element of $S$ strictly precedes $x$, as $x$ is in $T$.

Thus there is an order isomorphism from an initial segment of $T$ to all of $S$.

The cases are distinct by No Isomorphism from Woset to Initial Segment.

Construction of $\Omega$ without using choice
Munkres supplementary exercise 1.8

By the axiom of powers, there exists the power set $\mathcal P \left({\N}\right)$.

By Power Set of Natural Numbers Not Countable, this set is uncountable.

We construct a well-ordering $\left({P \left({\N}\right), \preccurlyeq}\right)$ that has the desired defining properties of $\Omega$.

Proof
Let $<$ denote the usual strict ordering on $\N$.

From the Well-Ordering Principle, $<$ is a well-ordering.

Denote:


 * $\mathcal A = \left\{ { \left({A,\prec}\right) : A \in\mathcal P(\N) }\right\}$

That is, the set of ordered pairs, such that:


 * the first coordinate is a (possibly empty) subset of $\N$


 * the second coordinate is any well-ordering on $A$.

We observe that there is at least one pair of this form for each $A$, taking $\prec \, = \, <$.

Define the relation:


 * $(A, <) \sim (A',<')$




 * $(A, <)$ is order isomorphic to $(A',<')$.

By Order Isomorphism is Equivalence Relation, $\sim$ is an equivalence relation.

Let $E$ be the set of all equivalence classes $\left[\!\left[{\left({A,<}\right)}\right]\!\right]$ defined by $\sim$ imposed on $\mathcal P(\N)$

Define:


 * $\left[\!\left[{\left({A,<_A}\right)}\right]\!\right] \ll \left[\!\left[{\left({B,<_B}\right)}\right]\!\right]$


 * $(A, <_A)$ is order isomorphic to an initial segment of $(B,<_B)$.

We claim that $\left({E,\ll}\right) = \Omega$.

Steps of proof:

$(a.1)$: this is well-defined

$(a.2)$: this is an ordering relation

$(a.2)$ $E$ has a smallest element $\left[\!\left[{\left({\varnothing,\varnothing}\right)}\right]\!\right]$