Can $[2,3[$ be considered an open interval? What about $]-\infty, 1]$? Can $[2,3[$ be considered an open interval? What about $]-\infty, 1]$?
Is there a definition of what exactly an open interval is?
Thank you.
 A: No, neither is open.
Yes, there is precise definition of what it means to be open: for every $x \in A$, there is a neighborhood $U$ of $x$ such that $U \subset A$. In this case, there are no such neighborhoods of $2$ or $1$.
A: One possible definition would be


*

*An interval is subset $I\subseteq\mathbb R$ such that whenever $a<b<c$ and $a\in I$ and $c\in I$ we also have $b\in I$.


*An interval $I$ is open if it doesn't contain any endpoint. More precisely: for every $b\in I$ there must be $a\in I$ and $c\in I$ with $a<b<c$.

This implies that the open intervals are those of the form $(a,b)$ or $(-\infty,b)$ or $(a,\infty)$ or $\mathbb R$ itself.
A: An interval $I$ is open iff for every point $x\in I$, there existes an $\delta(x)>0$ such that $(x-\delta,x+\delta)\subset I$.
Hence, both intervals are not open. The previous condition is not fulfilled in $x=2$ for $[2,3)$ and $x=1$ for $(-\infty,1]$.
A: In the real numbers, an interval is open if they are of the form $(a,b)$, where $a<b$ can be any real numbers, and $\infty$ or $-\infty$. 
Some examples of open intervals:
$(1,2)$
$(-\infty,\infty)$
$(5,\infty)$
A: Let $a,b\in\mathbb{R}$ then the following intervals are
\begin{alignat}{2}
(a,b) &= \{x\mid a<x<b\} &&{}\text{ open}\\
(a,b] &= \{x\mid a<x\leq b\} &&{}\text{ half-open or neither open nor closed}\\
[a,b) &= \{x\mid a\leq x<b\} &&{}\text{ half-open or neither open nor closed}\\
[a,b] &= \{x\mid a\leq x\leq b\} &&{}\text{ closed}
\end{alignat}
