0x001 Topology

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).