Is $U(c):=\{]a,b[~\mid ~]a,b[ \text{ is neighbourhood of } c \}$ a set?

Is $U(c):=\{]a,b[~\mid ~]a,b[ \text{ is neighbourhood of } c \}$ a set?

$]a,b[$ is neighbourhood of $c \in \mathbb{R}$ if $c \in ]a,b[$.
Is $U(c):=\{]a,b[~\mid~]a,b[ \text{ is neighbourhood of } c \}$ a set?
Thanks in advance!!