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2018년 8월 1일 수요일

0.7 The Extended Real Number System

We briefly talk about materials in elementary real analysis. We will not prove those properties (see elementary real analysis textbook such as Principles of Mathematical Analysis for reference).
We form the extended real number system by adding two elements -\infty and \infty on \mathbb{R}; \bar{\mathbb{R}} = \mathbb{R} \cup \left\{ -\infty, \infty \right\} Also we extend the usual ordering \leq on \mathbb{R} by defining -\infty < x < \infty, (\forall x \in \mathbb{R})

0.6 More about Well Ordered Sets

We discuss more about Well Ordered sets.
If X is well ordered and A \subset X, then \cup_{x \in A} I_{x} is either initial segment or X itself.
Let J = \cup_{x \in A} I_{x}. If J = X, nothing more to prove. If J \neq X, let b = \inf(X \setminus J). If y < b for y \in X, by construction, y \in J. This means I_{b} \subset J. Now pick y \in J. Then y \in I_{x} for some x \in A. If y \not\in I_{b}, then y \geq b gives b \leq y < x, in turn b \in I_{x} \implies b \in J; contradiction. So y \in I_{b}. Thus, J = I_{b}.
This proposition states that any union of initial segments are also an initial segment (including whole set) when the whole set is well ordered.
If X and Y are nonempty and well ordered, then either
  1. X is order isomorphic to Y
  2. X is order isomorphic to an initial segment of Y
  3. Y is order isomorphic to an initial segment of X
Consider the collection \mathcal{F} of order isomorphisms whose domains are initial segments of X or X itself and whose ranges are initial segments of Y or Y itself. \mathcal{F} is nonempty since f: \left\{ \inf X \right\} \rightarrow \left\{ \inf Y \right\} is a memeber of \mathcal{F}. Also \mathcal{F} is partially ordered by set inclusion if functions are viewed as subsets of X \times Y. By similar reasoning used in Section 0.4, Proposition 1-2, if S is nonempty linearly ordered subset of \mathcal{F}, then f^{\ast} = \bigcup_{f \in S} f is well defined function and upper bound of S. By Zorn's Lemma, \mathcal{F} has a maximal element; say g with domain A and range B. If A = I_{x} and B = I_{y} for some x \in X and y \in Y, then we can extend g by \bar{g}: A \cup \left\{ x \right\} \rightarrow Y,\ \bar{g}(a) = \begin{cases} g(a) & (a \in A) \\ y & (a = x) \end{cases} because I_{x} \cup \left\{ x \right\}, I_{y} \cup \left\{ y \right\} are again initial segments (including whole set X or Y). This contradicts to maximality of g. Hence either A = X or B = Y, and the result follows.
Well Ordering principle gives us the way to construct weird uncountable sets.
There is an uncountable well ordered set \Omega such that I_{x} is countable for every x \in \Omega. If \Omega' is another set with the same properties, then \Omega and \Omega' are order isomorphic.
First by well ordering principle, uncountable well ordered set \Omega_{0} exists. Then define A = \left\{ t \in \Omega_{0}: I_{t} \text{ is uncountable } \right\} If \Omega_{0} satisfies the desired property, then nothing more to prove. If not, A has minimal element a by well ordering principle. Then the set \Omega = I_{a} would be the desired set, because for any x \in \Omega, x \not\in A, so that I_{x} is countable. Any other \Omega' satisfying this property is order isomorphic to \Omega itself, because any initial segments of \Omega and \Omega' are countable. (See Proposition 2).
Every nonempty countable subset of \Omega has an upper bound.
Let A \subset \Omega be countable and nonempty. \cup_{x \in A} I_{x} is countable union of countable sets, so it is countable. By Proposition 1, it should be equal to I_{y} for some y \in \Omega. Then for any a \in A, I_{a} \subset I_{y} \implies a \leq y which is equivalent to saying y is an upper bound of A.
The set \Omega is called a set of countable ordinals. We usually define \Omega^{\ast} = \Omega \cup \left\{ \omega_{1} \right\} and extend the well ordering by declaring x < \omega_{1},\ (\forall x \in \Omega) \omega_{1} is called the first uncountable ordinal. We won't get further deep into theory of ordinals.

0.7 The Extended Real Number System

We briefly talk about materials in elementary real analysis. We will not prove those properties (see elementary real analysis textbook such ...