03 - Open and Closed

Open Ball

Let $r>0$ and $x\in {\mathbb{R}}^2$ then

$$B(x,r):=\{y\in {\mathbb{R}}^2:|x-y|<r\}$$

Closed Ball

Let $r>0$ and $x\in {\mathbb{R}}^2$ then

$$\overline{B}(x,r):=\{y\in {\mathbb{R}}^2:|x-y|\le r\}$$

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Open Set

Let set $U\subseteq {\mathbb{R}}^2$ be open if

$$\forall x\in U:\exists r>0\text{ such that }B(x,r)\subseteq U$$

Closed Set

Let set $U\subseteq {\mathbb{R}}^2$ be closed if

$$\text{complement }{\mathbb{R}}^2\setminus U\text{ is open}$$

Open Neighbourhood

Given $a\in {\mathbb{R}}^2$ then

Any open set $U\subseteq {\mathbb{R}}^2$ containing $a$ is an open neighbourhood of $a$

Topology in ${\mathbb{R}}^2$

The collection of all open sets in ${\mathbb{R}}^2$