Definition:Open Set/Metric Space

Definition
Let $M = \left({A, d}\right)$ be a metric space.

Let $U \subseteq M$.

Then $U$ can be described as iff:
 * an open set in $M$
 * open in $M$
 * a $d$-open set
 * $d$-open
 * $\forall y \in U: \exists \epsilon \left({y}\right) > 0: N_{\epsilon \left({y}\right)} \left({y}\right) \subseteq U$

That is, for every point $y$ in $U$, we can find an $\epsilon > 0$, dependent on that $y$, such that the open $\epsilon$-ball of that point lies entirely inside $U$.

Another way of saying the same thing is that one can not get out of $U$ by moving an arbitrarily small distance from any point in $U$.

It is important that the necessary value of $\epsilon$ is allowed to be different for each $y$.

Open Sets vs. Open Balls
It follows from Open Ball of Point Inside Open Ball that every open ball is an open set.

However, not every open set is an open ball.

For example, let $U \subseteq \R^2: U = \left\{{\left({x_1, x_2}\right): a < x_1 < b, c < x_2 < d}\right\}$.

Given $x = \left({x_1, x_2}\right) \in U$, it is easy to show that $B_{\epsilon} \left({x}\right) \subseteq U$ when $\epsilon = \min \left\{{x_1 - a, b - x_1, x_2 - c, d - x_2}\right\}$:


 * NeighborhoodInOpenSet.png

So $U$ is open in $M$, but it is not an open ball.

Open Set in Pseudometric Space
Let $P = \left({A, d}\right)$ be a pseudometric space.

An open set in $P$ is defined in exactly the same way as for a metric space:

$U$ is an open set in $P$ iff:
 * $\forall y \in U: \exists \epsilon \left({y}\right) > 0: B_{\epsilon \left({y}\right)} \left({y}\right) \subseteq U$

where $B_{\epsilon \left({y}\right)}$ is the open $\epsilon$-ball of $y$.