Topology

Definition: Topology

Topology by James Munkres, p.78 A topology on a set $X$ (topology space) is a collection $\mathcal{T}$ of subsets of $X$ having the following properties

1. $\varnothing$ and $X$ are in $\mathcal{T}$ (are open)
2. closure under finite intersections $\bigcap$
3. closure under arbitrary unions $\bigcup$

This collection $\mathcal{T}$ is the open sets, every element in this set is a open set

Question

Why do we only regulate that the number of intersections must be finite?

Simple Examples

Trivial Topology: $\mathcal{T}=\{\varnothing ,X\}$ Discrete Topology: $\mathcal{T}$ is generated by every element in $X$

Let $X=\{1,2,3\}$ $\mathcal{T}=\{\varnothing ,\{1\},\{1,2\},\{1,2,3\}\}$, $\mathcal{T}$ is a topology

Metric Topology

NOTE

This works only on metric space (where a distance function can be defined)

Consider a metric space $(M,d)$ ($d$ is the distance function) Let $\mathcal{T}$ be the collection of open sets in the usual metric space sense: A set $A$ is open (or in $\mathcal{T}$) if for every $x\in A$ there is an $\epsilon >0$ such that $B_{\epsilon }(x)\subseteq A$

NOTE

$B_{\epsilon }(x)$ is the ball set $\{y|y\in M,d(x,y)<\epsilon \}$

Think of a open set in this topology as a shape without boundary

Notation: Neighborhood

Topology by James Munkres, p.98

$U$ is an open set containing $x$

is the same as

$U$ is a neighborhood of $x$

NOTE

U is an open set containing x

Definition: Closed Set

Let $X$ be a topological space, a subset $F\subseteq X$ is closed if its complement $X\setminus F$ is open (or is in $\mathcal{T}$) Closed set is dual to open set under complementation, not under logical negation

NOTE

Note that a subset of $X$ can be open and closed at the same time or neither open nor closed

Definition: basic concepts

closure

A closure of $A$ in $X$ is the set

The closure $\bar{A}$ is the smallest closed set that contains $A$

interior

A interior of $A$ in $X$ is the set

The closure $\mathring{A}$ is the biggest open set that is contained in $A$. Or $Int(A)$

exterior

The exterior of $A$ is $\text{Ext}(A)=X\setminus \bar{A}$, and it is open

boundary

The boundary of $A$ is $\partial A\stackrel{def}{=}X\setminus (\mathring{A}\cup \text{Ext}(A))$, and it is closed

Relations

Note that we have $\bar{A}=\mathring{A}\cup \partial A=A\cup \partial A$