Is there a unnecessary part in this definition of "not connected set" in complex plane?

Question

In the book "Introduction to complex analysis" - Junjiro Noguchi, there is a definitoin of not connected set in complex plane as follows

Here, $A$ is a subset of complex plane. I think the part $\cap A$ in $U_1\cap U_2\cap A=\emptyset$ is not necessary. Am I wrong?

Answer 1

The definition is made so that it works in a general topology setting: $A$ in the subspace topology is a disconnected space. In a metric (or more generally a herditarily normal) space we get an equivalent definition if you omit the $\cap A$ in the final clause, see this question and its answers, but also note that it is easier to show disconnectedness using the stated definition, as we only have to ensure that $U_1$ and $U_2$ do not intersect in $A$, instead of at all. So I'd stick to this definition anyway, especially because it's also valid in any topological space you will encounter later.

Answer 2

Not an answer, just illustrating a point that is too long for a comment.

Let $X=\{a,b,c\}$ and $\tau=\{ \emptyset, \{a\}, \{a,b\}, \{a,c\}, X \}$ and let $A= \{b,c\}$.

With the definition above, $\{a,b\}$ and $\{a,c\}$ separate $A$.

If you omit the $\cap A$, then $A$ is non not connected.