Describe a neighborhood of a given interior point

Describe a neighborhood of a given interior point

Before I ask the question, I would like to define some terms.
A point $z_0$ is said to be an interior point of a subset $S$ of the
complex plane if there exists some neighborhood of $z_0$ that lies
entirely within $S$. If every point $z$ of a set $S$ is an interior point,
then $S$ is said to be an open set. For example, the inequality $\Re(z) >
1$ defines a right half-plane, which is an open set. All complex numbers
$z = x + iy$ for which $x > 1$ are in this set.
But does not it depend on the value of $y$? Suppose $z_0=1.1+2i$. How can
we describe a neighborhood in this case? Clearly the length between $z_0$
and some points should be less than some value, but what is this value?
Please help me.