Unions and Interesctions

Symbols

We have $\cup$ represent union and $\cap$ as intersection.

To distinguish countable and uncountable index, we have $\alpha$ as uncountable index.

DeMorgan's Law

$(\cup_\alpha \mathbb{E}_\alpha)^C = \cap_\alpha \mathbb{E}_\alpha ^C$

1. Arbitary union of open sets is open.

proof: $\forall x \in \cup_\alpha \mathbb{E}_\alpha$ ( $\mathbb{E}_\alpha$ is open)

$\Rightarrow x \in \mathbb{E}_\alpha$ for some $\alpha$.

$\Rightarrow x$ is an interior point to $\mathbb{E}_\alpha$,

$\Rightarrow x$ is an interior point to $\cup_\alpha \mathbb{E}_\alpha$

2. Arbitary intersection of closed sets is closed.

proof: $\forall x \in \cap_\alpha \mathbb{E}_\alpha$ ( $\mathbb{E}_\alpha$ is closed)

$\Rightarrow \forall \alpha, x \in \mathbb{E}_\alpha$.

$\Rightarrow x$ is an interior or limit point for all $\mathbb{E}_\alpha$,

$\Rightarrow x$ is an interior or limit point to $\cap_\alpha \mathbb{E}_\alpha$

3. Finite intersection of open sets is open.

4. Finite union of closed sets is closed.

Dense Sets

$\mathbb{E}$ is dense in metric $\mathbb{X}$ if every point of $\mathbb{X}$ is a limit point of $\mathbb{E}$ or belongs to $\mathbb{X}$

For example, $\mathbb{Q}$ is dense in $\mathbb{R}$.