Contents Index

## Topological Space

### Topology

**Definition (topology)** Let $X$ be a non-empty set. A set $\mathcal{T}$ of subsets of $X$ is said to be topology on $X$ iff

- $X \in \mathcal{T}, \emptyset \in \mathcal{T}$
- union of any number of sets of $\mathcal{T}$ belongs to $\mathcal{T}$
- interaction of any two sets of $\mathcal{T}$ belongs to $\mathcal{T}$

The pair $(X, \mathcal{T})$ are called topological space

**Definition (discrete, indiscrete topology)** $2^X$ is the discrete topology of $X$, $\{ \emptyset, X \}$ is the indiscrete topology of $X$.

**Definition (finer, coarser)** Suppose $\mathcal{T}, \mathcal{T’}$ are two different topologies on a given set $X$, if $\mathcal{T} \subset \mathcal{T’}$, then $\mathcal{T}$ is a coarser and $\mathcal{T’}$ is a finer topology

**Definition (open set)** $U \ subset X$ is an open set iff $U \in \mathcal{T}$

## Metric Space

**Definition (Metric spaces)** A metric space $(X,d)$ is a space $X$ of objects, together with a distance function or metric $d: X \times X \to [ 0, \infty ]$, which associates to each pair $x,y$ of points in $X$ a non-negative real number $d(x,y) \geq 0$. Furthermore, the metric must satisfy the following four axioms:

- $(\forall{x \in X}) d(x,x) =0$
- $(\forall{x, y \in X, x \neq y}) d(x,y) > 0$
- $(\forall{x, y \in X}) d(x,y) = d(y,x)$
- $(\forall{x,y,z \in X}) d(x,z) \leq d(x,y)+d(y,z)$

**Definition (Convergence)** Let $(X,d)$ be a metric space and let $(x^n)_{n=m}^{\infty}$ be a sequence of points in $X$, let $x \in X$. We say that $(x^n)_{n=m}^{\infty}$ converges to $x$ with respect to the metric $d$ iff $\lim_{n \to \infty} d(x^n, x)$ exists and is equal to 0

**Example ($l^{\infty}$ metric) **

$$d_{l^{\infty}}((x_1, x_2, …, x_n), (y_1, y_2, …, y_n)) := sup\{ |x_i – y_i|: 1 \leq i \leq n \}$$

**Example (discrete metric)** Let $X$ be an arbitrary set, the discrete metric is defined by $d_{disc}(x,y) := 0$ if $x=y$ otherwise $d_{disc}(x,y) = 1$

**Proposition (equivalence of $l^1, l^2, l^{\infty}$)** Let $R^n$ be a Euclidean space, $(x^{(k)})_{k=m}^{\infty}$ be a sequence of points in $R^n$. We write $(x^{(k)}=(x_1^{(k)}, x_2^{(k)}, …, x_n^{(k)})$. Let $x=(x_1, x_2, …, x_n)$ be a point in $R^n$. Then the following four statements are equivalenet

- $(x^{(k)})_{k=m}^{\infty}$ converges to $x$ with respect to $l^1$
- $(x^{(k)})_{k=m}^{\infty}$ converges to $x$ with respect to $l^2$
- $(x^{(k)})_{k=m}^{\infty}$ converges to $x$ with respect to $l^{\infty}$
- for all $1 \leq j \leq n$, the sequence $(x_j^{(k)})_{k=m}^{\infty}$ converges to $x_j$

## Open and Closed

### Topology of R

**Definition (open)** A set $O \subset R$ is open if all points $a \in O$ has a $\epsilon$ neighborhood $V_{\epsilon}(a) \subset O$

**Lemma (Properties of open set)**

- any union of open sets is open (up to uncountable union)
- a finite intersection of open sets is open (because min radius > 0)

The infinite intersection might not be open (e.g: sup might reduce the radius to 0)

**Proposition (characterization of open subsets) **open subset of R is countable disjoint union of open intervals

**Definition (limit point)** A point $x$ is a limit point of a set $A$ if every $\epsilon$ neighborhood of $x$ intersects the set $A$ at some point other than $x$

This essentially means that $x$ can be a limit of sequence on $A$.

Note that limit point $x$ might not belong to $A$

**Definition (isolated point)** A point $x \in A$ is an isolated point of $A$ if it is not a limit point of $A$

Note that an isolated point must belong to $A$. Isolation means that there is a neighborhood where the point can be detached from the remaining of the other points.

**Definition (closed)** A set $F \subset R$ if it contains its limit points.

Closed set essentially means the set is closed under limit operations

**Lemma (properties of closed set)**

- finite union of closed set is closed
- any intersection of closed set is closed (up to uncountable intersection)

Infinite union of closed set might not be closed (e.g: $\cup_{n} [1/n, 1]$ is not closed as 0 does not belong to it)

**Examples (closed set)**

- cantor set

**Definition (adherent point, closure point)** $x$ is called an adherent point of $A$ if every neighborhood of $x$ contains at least one point of $A$.

Adherent point is either limit point or isolated point

**Definition (closure)** A closure of $A$ is a set $\bar{A}$ combining all limit points $L$ with $A$

$$\bar{A} = A \cup L$$

Closure is a closed set and is the smallest closed set containing $A$

**Theorem** $O$ is open $\iff$ $O^c$ is closed

**Definition (perfect)** A set $P \subset R$ is *perfect* if it is closed and contains no isolated points

A nonempty perfect set is uncountable

### Topology of Metric Space

**Definition (Ball)** The ball $B_{(X,d)} (x_0, r)$ in a metric space where $x_0 \in X, r >0$ is defined to be the set

$$B_{(X,d)} (x_0, r) := \{ x \in X: d(x, x_0) \in r \}$$

**Definition (interior, boundary, exterior) **

- $x_0 \in X$ is an interior iff $(\exists r >0) B(x_0, r) \subseteq E$
- $x_0 \in X$ is an exterior iff $(\exists r >0) B(x_0, r) \cap E = \emptyset$
- $x_0$ is a boundary point if it is neither an interior or exterior

**Definition (adherent point) **$x_0$ is an adherent point of $E \subseteq X$ iff

$$(\forall r > 0) B(x_0, r) \cap E \neq \emptyset$$

Note: adherent point does not equal to limit point as it allows isolated point, but limit point need a sequence (different from itself) to approach it

**Definition (closure)** The set of all adherent point of $E$ is called the *closure* of $E$ and is denoted $\bar{E}$

**Definition (open, closed) **

- $E$ is closed if it contains all of its boundary points
- $E$ is open if it contains none of its boundary points

## Connectedness

**Definition (connected space) **Let $X$ be a topological space, A *separation* of $X$ is a pair $U,V$ of disjoint open subsets of $X$ whose union is $X$. The space is said to be connected if there is no separation of $X$.

**Definition (separated) **Two nonempty sets $A,B \subset R$ are *separated* if $\bar{A} \cap B$ and $A \cap \bar{B}$ are both empty

**Definition (disconnected)** A set $E \subset R$ is *disconnected* if it can be written as $E = A \cup B$ where $A,B$ are separated. If it is not disconnected, it is called connected

**Lemma (connected)** A set is connected if and only if for all disjoint sets $A,B$ where $E = A \cap B$, there is always a convergent sequence from one side can limit into the other one.

**Lemma** A set $E \subset R$ is connected iff whenever $a < c < b$ with $a,b\in E$ it follows that $c \in E$

**Definition (connectedness in metric space) **Let $(X, d)$ be a metric space. We say that $X$ is disconnected iff there exist disjoint non-empty open sets $V, W$ such that $X=V \cup W$

## Compactness

compactness is a concept generalizing a bounded closed set.

**Definition (sequential compactness)** A set $K \subset R$ is sequential-compact if every sequence in $K$ has a subsequence that converges to some limit in $K$.

**Definition (compactness) **A set $K \subset R$ is compact if every open cover for $K$ has a finite subcover

**Theorem (Heine-Borel) **Following statements are equivalent

- $K$ is sequential compact
- $K$ is compact
- $K$ is closed and bounded

## Reference

[1] Abbott, Stephen. *Understanding analysis*. Vol. 2. New York: Springer, 2001.

[2] Tao, Terrence. “Analysis (Volume 1).” *Hindustan Book Agency* (2006).