Is an open domain in the complex plane a generalization of an open interval on the real line? An open domain in the complex plain is thought of as a generalization of an open interval on the real line.  Can this be proven at least for some specific domain(s) and some type(s) of holomorphic functions?
As an attempt to doing this, I considered the open line segment AB, in the complex on defined by :
    $$a < x < b ;  a > 0 ; y = t$$
And the open (rectangular) domain CDEF, around the segment AB, and defined by:
    $$a  <  x <  b ;   y  = t – e  ;  e  > 0$$ and:
$$ a  <  x <  b ;   y  = t + e  $$
So that the length of sides CE and DF is equal to 2e 
Now when the value e goes to zero, the sides CE and DF reduce to the points A and B respectively, and the interior of the domain CDEF reduces to the segment AB, so that the interior of the domain CDEF is confounded with the border of the segment AB. This result introduces a controversy: is the open segment AB a border, and interior to the domain limit of CDEF? If it’s the interior, then according to the maximum modulus principle and by continuity, any holomorphic function defined in the open domain is zero, which is not true, unless the function is zero in the domain CDEF.
This makes the generalization of an open interval on the real line not possible. 
Any thoughts on when (for which type of functions and type of domain) the generalization is true would be useful?
Thank you
 A: It seems like what is confusing is that there is no homeomorphism (continuous bijection with a continuous inverse) between an interval $\{ x+ic\ |\ a < x<b, c=constant  \}$ (considered now as a subset of $\mathbb{C}$) and a domain in $\mathbb{C}$. 
This is correct, as you have shown -- an "open interval" of the real line is no longer open in the topology of $\mathbb{R}^2$ -- it is only open in the topology of $\mathbb{R}$. In other words, your confusion is justified, inasmuch as open intervals are not "open" when considered in $\mathbb{C}$.
Take note: the terms "open" and "connected" are not independent of the space you are working with. In fact, they can have different meanings for the same space $X$. The terms "open" and "connected" only have meaning with reference/respect to a particular choice of structure on the space, called a "topology". Thus we consider pairs $(X,\tau)$, where $\tau$ is a topology for $X$ -- thus the terms "open" and "connected" for subsets of $X$ are always defined with respect to a choice of $\tau$. Potentially, we could have two different topologies for $X$, call them $\tau_1, \tau_2$. Then it could be possible for a subset $Y \subseteq X$ to be open with respect to $\tau_1$ but not open with respect to $\tau_2$, for example. Hence it is even more plausible that for two different spaces $X_1, X_2$, for example $X_1=\mathbb{R}, X_2 = \mathbb{C}$, and for choices of topologies on them, $(X_1, \lambda), (X_2,\mu)$, that a subset $Y \subseteq X_1, Y\subseteq X_2$ of both spaces could be open with respect to $\lambda$ but not with respect to $\mu$, or open with respect to $\mu$ but not with respect to $\lambda$. Even when $X_1 \subseteq X_2$.
open intervals (a,b) -- as subsets of $\mathbb{R}$, they are the only open (with respect to the topology of $\mathbb{R}$) and connected (with respect to the topology of $\mathbb{R}$) sets.
As subsets of $\mathbb{C}$, open intervals $\{ x+ic\ |\ a<x<b, c=constant \}$ are connected (with respect to the topology of $\mathbb{C}$) but not open (with respect to the topology of $\mathbb{C}$).
domains in the complex plane -- as subsets of $\mathbb{C}$, they are the only open (with respect to the topology of $\mathbb{C}$) and connected (with respect to the topology of $\mathbb{C}$) sets.
Note: when I say "the topology on $\mathbb{R}$" or "the topology on $\mathbb{C}$" I mean "the standard topology on $\mathbb{R}$" respectively "the standard topology on $\mathbb{C}$" -- technically there are other possible choices for topologies on those sets. Another name for the standard topologies on these sets is the "Euclidean topology" (because it is the standard choice of topology for "Euclidean space").
What I will ask you to consider is this -- given any open domain in $\mathbb{C}$, for example an open rectangle, but even an open circle or triangle or ellipse, its intersection with the real line $\mathbb{R}\subset \mathbb{C}$, if not the empty set, is an open interval in $\mathbb{R}$. This is not a coincidence.
Right now I am probably confusing you more than I am helping you, so I will ask you to read about subspace topologies (e.g. the topology of $\mathbb{R}$) and ambient space topologies (e.g. the topology of $\mathbb{C}$) here on Wikipedia: https://en.wikipedia.org/wiki/Subspace_topology.
Also another relevant fact: the Cartesian product of two open intervals in $\mathbb{R}$ is an open rectangle in $\mathbb{R}^2$ -- this is also not a coincidence. I don't know how to explain this fact in a general setting without getting too involved with topology, see for example here or here.
