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Math4121 Lecture 17
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Countability
Theorem: \mathbb{R} is uncountable
We denote the cardinality of \mathbb{N} be \aleph_0
We denote the cardinality of \mathbb{R} be \mathfrak{c}
Continuum Hypothesis:
If there a cardinality between
\aleph_0and\mathfrak{c}
Power set
Definition: Power set
Given a set S, the power set of S, denoted \mathscr{P}(S) or 2^S, is the collection of all subsets of S.
Theorem 3.10 (Cantor's Theorem)
Cardinality of 2^S is not equal to the cardinality of S.
Proof:
Assume they have the same cardinality, then \exists \psi: S \to 2^X which is one-to-one and onto. (this function returns a subset of S)
T=\{a\in S:a\notin \psi (a)\}\subseteq S
Thus, \exists b\in S such that \psi(b)=T.
If b\in T, then by definition of T, b \notin \psi(b), but \psi(b) = T, which is a contradiction. So b\notin T.
If b \notin T, then b \in \psi(b), which is also a contradiction since b\in T. Therefore, 2^S cannot have the same cardinality as S.
QED
Back to Hankel's Conjecture
T=\bigcup_{n=1}^\infty \left(a_n-\frac{\epsilon}{2^{n+1}},a_n+\frac{\epsilon}{2^{n+1}}\right)
is small
What is the structure of S=[0,1]\setminus T? (or Sparse)
- Cardinality (countable)
- Topologically (not dense)
- Measure, for now meaning small or zero outer content.
Chapter 4: Nowhere Dense Sets and the Problem with the Fundamental Theorem of Calculus
Nowhere Dense Sets
Definition: Nowhere Dense Set
A set S is nowhere dense if there are no open intervals in which S is dense.
Corollary: A set is nowhere dense if and only if S contains no open intervals
S' contains no open intervals