Number Systems and Basic Set Theory

Let’s just skip this section first.

Basic Topology

Countable Sets

::: thm
Theorem 1. If there is a surjection $f: A \rightarrow B$, we have
$|A| \geq |B|$. If there is an injection $f: A \rightarrow B$, we have
$|A| \leq |B|$.
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::: dfn
Definition 2. (Equivalence between sets) If there is a bijection
form set $A$ to set $B$, we say they have the same cardinality number.
In short, they are equivalent. We write $A \thicksim B$.
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::: dfn
Definition 3. [Finite, Countable]{style=”color: RED”}

Let $S_n$ represents set {1, 2, 3, ..., n}. If set $A$ satisfies
$A \thicksim Sn$ for some $n \in \mathbb{N}$, $A$ is finite. If
$A$ is not finite, $A$ is infinite.
Let $S$ represents the set of all positive integers. If set $A$
satisfies $A \thicksim S$, $A$ is countable. A set is
uncountable if it is neither finite nor countable. [^1]
$A$ is at most countable if it is countable or finite.
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::: rmk
Remark 4. A finite set can’t be equivalent to one of its proper
subsets, but some infinite sets can.
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::: thm
Theorem 5. Every infinite subset of a countable set $A$ is
countable.
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::: thm
Theorem 6. An at most countable union of at most countable sets are
countable.
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::: cor
Corollary 7. The set of rational numbers are countable, but the set
of real number are not.
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::: cor
Corollary 8. The set of infinite sequences
$A = {a_0, a_1, a_2, …, a_n, …}$ is uncountable.
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::: cor
Corollary 9.
$\mathbb{R} \thicksim \mathbb{R}^{2} \thicksim \mathbb{R}^{n}$ for all
$n \in \mathbb{N}$.
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Metric Spaces

::: dfn
Definition 10. [Metric space, distance
function]{style=”color: RED”}

A combination of a set $X$ and a map
$d: X \times X \rightarrow \mathbb{R}$, which is written as
$(X, d)$, is called a metric space if for $\forall p,:q\in X$:
1. $d(p,:q) > 0$, if $p \neq q$; $d(p,:q) = 0$;
2. $d(p,:q) = d(q,:p)$;
3. $d(p,:q) \leq d(p,:r) + d(r,:q)$ for $\forall r \in X$.
Every function $d$ such that satisfies the three properties are
called distance function.
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::: dfn
Definition 11. [Segment, Interval]{style=”color: RED”}

The set of rational numbers such that $a < x < b$ for any given
$a, b\in\mathbb{R};s.t.;a < b$ is called the segment $(a, b)$.
The set of rational numbers such that $a \leq x \leq b$ for any
given $a, b\in\mathbb{R};s.t.;a \leq b$ is called the interval
$(a, b)$.
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::: dfn
Definition 12. [Neighborhood, Limit point, Interior
point]{style=”color: RED”}

For any given metric space $(X, d)$ and set $E \subseteq X$, we have
following definitions:

A neighborhood of a point $p$ with radius $r>0$ is a set
$N_{r}(p) = {q \in X: d(p,:q) < r}$.
A point $p$ is a limit point of set $E$ if
$\forall r \in R, r > 0$, we have
$(N_{r}(p) \setminus {p}) \cap E \neq \emptyset$. The set of all
limit points of set $E$ is written as $E’$.
A point $p$ is a interior point of set $E$ if
$\exists r \in R, r > 0;s.t.;N_{r}(p) \subseteq E$.
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::: dfn
Definition 13. [Closed, Open, Closure]{style=”color: RED”}

For any given metric space $(X, d)$ and set $E \subseteq X$, we have
following definitions:

Set $E$ is open if all limit points of $E$ is in $E$. That is,
$E’ \subseteq E$.
Set $E$ is closed if every point of $E$ is an interior point of
$E$.
The closure ${}\mkern 3mu\overline{\mkern-3muE}$ of a set $E$ is the
union of its limit points and itself. That is,
${}\mkern 3mu\overline{\mkern-3muE} = E \cup E’$.
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::: dfn
Definition 14. [Complement, Perfect, Bounded,
Dense]{style=”color: RED”}

For any given metric space $(X, d)$ and set $E \subseteq X$, we have
following definitions:

The complement of $E$ is $E^{c} = X \setminus E$.
$E$ is perfect if $E = E’$.
$E$ is bounded if there is a real number $r’$ and a point
$p \in X$ s.t. $E \subseteq N_{r’}(p)$.
$E$ is dense in $X$ if $X = {}\mkern 3mu\overline{\mkern-3muE}$.
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::: cor
Corollary 15. Every neighborhood is an open set.
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::: cor
Corollary 16. $\forall p \in E’, \forall r > 0$, set $E \cap N_r(p)$
is infinite. Therefore, a finite set has no limit point.
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::: thm
Theorem 17.
$(\bigcup_{\alpha}{E_{\alpha}})^{c} = \bigcap_{\alpha}(E_{\alpha}^{c})$.
That is, the complement of a union of sets is the intersection of the
complements of each set.
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::: thm
Theorem 18. A set $E$ is open if and only if its complement is
closed. $E$ is closed if and only if its complement is open.
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::: thm
Theorem 19. The relationship of the union/intersection of sets
between its openness and closedness:

For a finite collection of open sets, both its intersection and
union are open;
For a finite collection of closed sets, both its intersection and
union are closed;
For a infinite collection of open sets, only its union is guaranteed
to be open;
For a infinite collection of closed sets, only its intersection is
guaranteed to be closed.
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::: thm
Theorem 20. If $X$ is a metric space and $E \subset X$, then:

${}\mkern 3mu\overline{\mkern-3muE}$ is closed;
$E = {}\mkern 3mu\overline{\mkern-3muE}$ if and only if $E$ is
closed;
$\forall F \subset X$ such that $E \subset F$ and $F$ is closed,
${}\mkern 3mu\overline{\mkern-3muE} \subset F$.
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::: cor
Corollary 21. Let $E$ be a nonempty set of real numbers which is
bounded above, then sup $E \in {}\mkern 3mu\overline{\mkern-3muE}$,
andsup $E \in E$ if E is closed.
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::: dfn
Definition 22. [Open relative]{style=”color: RED”}

For any subset $E$ and $Y$ of metric space $X$ such that
$E \subset Y$, we say $E$ is open relative to $Y$ if for each
point $p \in E$, there is a radius $r > 0$ such that
$\forall q \in Y$ and $d(p, q) < r, q \in E$.
It’s clear to see that the definition is equivalent to there is an
open subset $G \subset X$ such that $E$ = $Y \cap G$.
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Compact

::: dfn
Definition 23. [Open cover, Subcover, Compact, Relatively
Compact]{style=”color: RED”}

An open cover of $E$ is a collection of open subsets
${G_{\alpha}}$ of $X$, such that
$E \subseteq \bigcup_{\alpha}G_{\alpha}$. That is, $E$ is
"covered" by ${G_{\alpha}}$.
A subcover of an open cover ${G_{\alpha}}$ is the subset of
${G_{\alpha_{i}}}$, while ${G_{\alpha_{i}}}$ still "cover" the
set $E$, that is, ${\alpha_{i}}\subset {\alpha}$ while
$E \subseteq \bigcup_{\alpha_{i}}G_{\alpha_{i}}$.
$E$ is compact in $X$ if for every open cover ${G_{\alpha}}$,
there is a $finite$ subcover.
Suppose $K \subset Y \subset X$, then $K$ is compact in $X$ if and
only if $K$ is compact in $Y$, which can also be written as $K$ is
compact relative to $Y$.
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::: thm
Theorem 24. Compact subset and metric space are closed and
bounded.
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::: thm
Theorem 25. Every closed subset of compact set is compact.
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::: cor
Corollary 26. If $A$ is closed and $B$ is compact, then $A \cap B$
is compact too.
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::: thm
Theorem 27. For any given collection of compact sets
${K_{\alpha}}$ such that for every finite subset
${K_{\alpha_{i}}}{i=1}^{n}$, its intersection
$\bigcap{i=1}^{n}K_{\alpha_{i}}$ is not empty, then
$\bigcap K_{\alpha}$ is not empty too.
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::: cor
Corollary 28. For a chain of nonempty compact sets in $X$ such that
$K_{n} \subset K_{n+1}$ is not empty for any $n \geq 1$, then
$\bigcap K_{n}$ is not empty too.
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::: thm
Theorem 29. For any compact set $K$, every infinite subset of $K$
has a limit point in $K$.
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[^1]: There are different definitions for countable. Some people say if
a set is finite, it is also countable, and infinite countable set is
"infinitely countable". In the following paragraphs, we will
mainly follow the definition from Rudin.

Study note of MATH 521 - Analysis I

Number Systems and Basic Set Theory

Basic Topology

Countable Sets

Metric Spaces

Compact