Definition:Open Set/Real Analysis

Definition

Real Numbers

Let $I \subseteq \R$ be a subset of the set of real numbers.

Then $I$ is open (in $\R$) if and only if:

$\forall x_0 \in I: \exists \epsilon \in \R_{>0}: \openint {x_0 - \epsilon} {x_0 + \epsilon} \subseteq I$

where $\openint {x_0 - \epsilon} {x_0 + \epsilon}$ is an open interval.

Note that $\epsilon$ may depend on $x_0$.

Real Euclidean Space

Let $n \ge 1$ be a natural number.

Let $U \subseteq \R^n$ be a subset.

Then $U$ is open (in $\R^n$) if and only if:

$\forall x \in U : \exists R \in \R_{>0}: \map B {x, R} \subset U$

where $\map B {x, R}$ denotes the open Euclidean ball of radius $R$ centered at $x$.

Also see

• By Open Sets in Real Number Line every open set $I \subseteq \R$ is a countable union of pairwise disjoint open intervals:

$\ds I = \bigcup_{n \mathop \in \N} \openint {a_n} {b_n} \subseteq \R$

There is believed to be a mistake here, possibly a typo. In particular: The above means the union of ALL real intervals, not the union of an arbitrary subset of them. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by reviewing it, and either correcting it or adding some explanatory material as to why you believe it is actually correct after all. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Mistake}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

• Definition:Closed Set (Real Analysis)