05 - Topological Properties

Path Connected

For every $a_0,a_1\in U$ there exists a path between $a_0$ and $a_1$

Simply-Connected

Let $U\subseteq {\mathbb{R}}^2$ then

$U$ is simply connected if

1. path-connected
2. For $a_0,a_1\in U$ then any two paths in $U$ between $a_0$ and $a_1$ is homotopic

Connected

Let $U\subseteq {\mathbb{R}}^2$ then

$U$ is connected if for any open sets $A,B\in {\mathbb{R}}^2$ with

$$U\subseteq A\cup B\text{ and }(A\cap B)\cap U=\varnothing$$

Then

$$U\subseteq A\text{ or }U\subseteq B$$

Compactness

Let $U\subseteq {\mathbb{R}}^2$ then

$U$ is connected if
Given any collection $\{V_i:i\in I\}$ of open sets with $U\subseteq {\bigcup }_{i\in I}{V}_i$
There exists a finite subset $J\subseteq I$ such that $U\subseteq {\bigcup }_{j\in J}{V}_j$

Path Connected Sets are always Connected

Open Connected Set implies Path Connected (in ${\mathbb{R}}^2$)

Bounded

$U\subseteq {\mathbb{R}}^2$ is bounded if

$$\exists x\in {\mathbb{R}}^2\text{ and }r>0\text{ such that}U\subseteq B(x,r)$$