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

• an open set in $M$
• open in $M$
• a $d$-open set
• $d$-open

iff:

$\forall y \in U: \exists \epsilon \left({y}\right) > 0: N_{\epsilon \left({y}\right)} \left({y}\right) \subseteq U$

That is, for every point $y$ in $U$, we can find an $\epsilon > 0$, dependent on that $y$, such that the $\epsilon$-neighborhood 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. Neighborhoods

It follows from Neighborhood of Point Inside Neighborhood that every neighborhood is an open set.

However, not every open set is a neighborhood.

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 $N_{\epsilon} \left({x}\right) \subseteq U$ when $\epsilon = \min \left\{{x_1 - a, b - x_1, x_2 - c, d - x_2}\right\}$:

So $U$ is open in $M$, but it is not a neighborhood.

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:

$\forall y \in U: \exists \epsilon \left({y}\right) > 0: N_{\epsilon \left({y}\right)} \left({y}\right) \subseteq U$

where $N_{\epsilon \left({y}\right)}$ is the $\epsilon$-neighborhood of $y$.

Sources

• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Definition $2.3.8$, Example $2.3.9$