= Open set
{title2=$U\in\tau$}
{wiki}
= Open subset
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= Open
{synonym}
An open set is a member of the <topology> on a <topological space>. In a <metric space>, a <subset> $U$ is an <open set> precisely when every $x\in U$ has an <open ball> $B(x,r)$ contained in $U$ for some $r>0$. For $U=(0,1)$, one can take $r=\tfrac12\min(x,1-x)$. At $x=0.2$, this gives $r=0.1$ and $B(x,r)=(0.1,0.3)\subset U$. The radius depends on the point; no uniform radius is required. https://ocw.mit.edu/courses/18-100b-real-analysis-spring-2025/resources/ocw_18100b-lec13-2025apr03_mp4/[MIT Real Analysis, Lecture 13]{ref}
Openness depends on the ambient <topological space>. In $A=[0,1]$ with its <subspace topology>, $[0,1/2)=A\cap(-1,1/2)$ is an <open set> in $A$, although it is not an <open set> in the <real line>. Similarly, the whole <closed interval> $A$ is both an <open set> and a <closed set> in itself. These relative <open sets> are what the <continuity> test uses for maps involving a <subset> with its <subspace topology>.
Arbitrary <intersections> of <open sets> need not be <open sets>: $\bigcap_{n\geq1}(-1/n,1/n)=\{0\}$, a <singleton> that is not an <open set> in the usual <real line>. Any point $x\ne0$ is excluded once $n>1/|x|$. This is why the <topology axiom> specifies finite <intersections>.
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