Definition:Open Set

Topology

Let $T = \left({X, \vartheta}\right)$ be a topological space.

Then the elements of $\vartheta$ are called the open sets of $T$.

Thus:

$U \in \vartheta$

and:

$U$ is open in $T$

are equivalent statements.

Metric Space

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 $y$.

Complex Analysis

Let $S \subseteq \C$ be a subset of the set of complex numbers.

Suppose that $\forall z_0 \in S: \exists \epsilon > 0: N_{\epsilon} \left({z_0}\right) \subseteq S$

where $N_{\epsilon} \left({z_0}\right)$ is the $\epsilon$-neighborhood of $z_0$ for some real $\epsilon > 0$.

Then $S$ is described as open (in $\C$).

Note that $\epsilon$ may depend on $z_0$.

Real Analysis

By Open Sets in Reals every open set $I \subseteq \R$ is a countable union of disjoint open intervals:

$\displaystyle I = \bigcup_{n \in \N} \left({a_n . . b_n}\right) \subseteq \R$