Finite Intersection of Open Sets of Normed Vector Space is Open

Theorem
Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space.

Let $U_1, U_2, \ldots, U_n$ be open in $M$.

Then $\displaystyle \bigcap_{i \mathop = 1}^n U_i$ is open in $M$.

That is, a finite intersection of open subsets is open.

Proof
Let $\displaystyle x \in \bigcap_{i \mathop = 1}^n U_i$.

For each $i \in \closedint 1 n$, we have $x \in U_i$.

Thus:
 * $\exists \epsilon_i > 0: \map {B_{\epsilon_i}} x \subseteq U_i$

where $\map {B_{\epsilon_i}} x$ is the open $\epsilon_i$-ball of $x$.

Let $\displaystyle \epsilon = \min_{i \mathop = 1}^n \set {\epsilon_i}$.

So:


 * $\epsilon > 0$.

Let $y \in \map {B_\epsilon} x$.

Then $\norm {x - y} < \epsilon$

Hence:


 * $\forall i \in \closedint 1 n : \norm {x - y} < \epsilon_i$

In other words:
 * $\map {B_\epsilon} x \subseteq \map {B_{\epsilon_i}} x \subseteq U_i$

for all $i \in \closedint 1 n$.

So:
 * $\displaystyle \map {B_\epsilon} x \subseteq \bigcap_{i \mathop = 1}^n U_i$

The result follows.

Also see

 * Finite Intersection of Open Sets is Open