Open set
For all real numbers $a$ and $b$ such that $a < b$, the interval of numbers strictly between $a$ and $b$ is called an open interval, and is written $(a, b)$.
Consider a set $U$ of points on the number line, in the plane, or in space. The set $U$ is open if every point of $U$ has a small ball around it that lies entirely inside $U$.
Consider a set $U$ in a metric space. The set $U$ is open if, for every point $x \in U$, there is an $\varepsilon > 0$ such that
\[B(x, \varepsilon) \subseteq U,\]where $B(x, \varepsilon)$ is the open ball of radius $\varepsilon$ around $x$.
Consider a set $X$ with a topology $\mathcal{T}$. A set $U \subseteq X$ is open if $U \in \mathcal{T}$.
Examples
Examples
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The empty set $\emptyset$
(definition)
High school
The empty set is open in $\reals$, because it has no points, so the condition on its points holds vacuously.
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$(0, 1)$
High school
The interval $(0, 1)$ is open in $\reals$, because every point of $(0, 1)$ has a small interval around it inside $(0, 1)$.
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The set of real numbers, $\reals$
High school
The set $\reals$ is open as a subset of itself, because every interval around a real number is inside $\reals$.
Non-examples
Things that may be mistaken for examples.
-
$[0, 1]$
High school
The interval $[0, 1]$ is not open in $\reals$, because every interval around $0$ contains negative numbers, which are not in $[0, 1]$.
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$[0, 1)$, as a subset of $\reals$
High school
The interval $[0, 1)$ is not open in $\reals$, because every interval around $0$ contains negative numbers.
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The Cantor set
Undergraduate
The Cantor set is not open, because it contains no interval of positive length, so no point of the Cantor set (such as $0$) has a small interval around it inside the Cantor set.
Properties
- Every union of open sets is open, and every intersection of finitely many open sets is open. An intersection of infinitely many open sets need not be open: the intersection of the intervals $(-1/n, 1/n)$ is $\setdef{0}$.
- A set is open if and only if it equals its interior, and if and only if it contains none of its boundary points.
Notes
Being open depends on the surrounding space: $[0, 1)$ is not open in $\reals$, but it is open as a subset of the space $[0, \infty)$.
Distinctions involving this term
- Topology vs. σ-algebra: Topologies and σ-algebras are both collections of subsets that are closed under some operations. A topology is closed under arbitrary unions and finite intersections, and describes nearness. A σ-algebra is closed under complements and countable unions, and describes what can be measured.