Basic Topology on Metric Spaces

Countable Sets

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 $|A| \leq |B|$.

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 $A \thicksim B$.

Definition 3. [Finite, Countable]

• Let $S_n$ represents set {1, 2, 3, ..., n}. If set $A$ satisfies $A \thicksim S_n$ 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.

Remark 4. A finite set can't be equivalent to one of its proper subsets, but some infinite sets can.

Theorem 5. Every infinite subset of a countable set $A$ is countable.

Theorem 6. An at most countable union of at most countable sets are countable.

Corollary 7. The set of rational numbers are countable, but the set of real number are not.

Corollary 8. The set of infinite sequences $A = \{a_0, a_1, a_2, ..., a_n, ...\}$ is uncountable.

Corollary 9. $\mathbb{R} \thicksim \mathbb{R}^{2} \thicksim \mathbb{R}^{n}$ for all $n \in \mathbb{N}$.

Metric Spaces

Definition 10. [Metric space, distance function]

• 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.

Definition 11. [Segment, Interval]

• 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 $(a, b)$.

Definition 12. [Neighborhood, Limit point, Interior point]

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 $\exists r \in R, r > 0\;s.t.\;N_{r}(p) \subseteq E$.

Definition 13. [Closed, Open, Closure]

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, ${}\mkern 3mu\overline{\mkern-3muE} = E \cup E'$.

Definition 14. [Complement, Perfect, Bounded, Dense]

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 $X = {}\mkern 3mu\overline{\mkern-3muE}$.

Corollary 15. Every neighborhood is an open set.

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.

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.

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.

Theorem 19. The relationship of the union/intersection of sets between its openness and closedness:

1. For a finite collection of open sets, both its intersection and union are open;

2. For a finite collection of closed sets, both its intersection and union are closed;

3. For a infinite collection of open sets, only its union is guaranteed to be open;

4. For a infinite collection of closed sets, only its intersection is guaranteed to be closed.

Theorem 20. If $X$ is a metric space and $E \subset X$, then:

1. ${}\mkern 3mu\overline{\mkern-3muE}$ is closed;

2. $E = {}\mkern 3mu\overline{\mkern-3muE}$ if and only if $E$ is closed;

3. $\forall F \subset X$ such that $E \subset F$ and $F$ is closed, ${}\mkern 3mu\overline{\mkern-3muE} \subset F$.

Corollary 21. Let $E$ be a nonempty set of real numbers which is bounded above, then sup $E \in {}\mkern 3mu\overline{\mkern-3muE}$, and sup $E \in E$ if E is closed.

Definition 22. [Open relative]

• 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 $E$ = $Y \cap G$.

Compact

Definition 23. [Open cover, Subcover, Compact, Relatively Compact]

• 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 $Y$.

Theorem 24. Compact subset and metric space are closed and bounded.

Theorem 25. Every closed subset of compact set is compact.

Corollary 26. If $A$ is closed and $B$ is compact, then $A \cap B$ is compact too.

Theorem 27. For any given collection of compact sets $\{K_{\alpha}\}$ such that for every finite subset $\{K _{\alpha _{i}}\} ^{n} _{i=1}$, its intersection $\bigcap ^{n}_{i=1}K _{\alpha _{i}}$ is not empty, then $\bigcap K_{\alpha}$ is not empty too.

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.

Theorem 29. For any compact set $K$, every infinite subset of $K$ has a limit point in $K$.

References

1. There are different definitions for countable. Some people say if a set is finite, it is also countable, and infinite countable set it "infinitely countable". In the following paragraphs, we will mainly follow the definition from Rudin. ↩