My text define a surface in this way:
let $A \subseteq \mathbb{R}^2$ be an open connected set, T a set such that $A \subseteq T \subseteq \mathrm{Cl}(A)$ and $r: T \to \mathbb{R}^3$ a continuous map, then the ordered pair $(r(T), r)$ is called a surface. Moreover it remarks that with this hypothesis A is equal to the interior of $T$.
I have two questions:
It is obvious that $A \subseteq \mathrm{Int}(T)$, but I can't convince myself of the other inclusion. For example, if $A = \mathbb{R}^2 \backslash ([0, +\infty) \times \{0\}) $ and $T = \mathbb{R}^2 \backslash ([2, +\infty) \times \{0\}) $ the hypothesis are verified (A is connected because it is a star domain, take (-1, 0) for example) but $(1,0) \in \mathrm{Int}(T)\backslash A$. Am I missing something?
It seems to me that the role of $A$ is to avoid 'weird' choice of $T$, for example we can't have a surface with domain $(\mathbb{Q} \cap[0,1])^2$, am I correct? are there more profound reasons?
Thanks in advance
This is the most bizarre definition of "surface" I've ever seen.
First of all, it's not generally true under these conditions that $A$ is equal to the interior of $T$. For example, $A$ could be the open unit disk with the origin removed, and $T$ could be the entire open unit disk. Then the interior of $T$ is $T$ itself, not $A$.
Second, this definition allows for a lot of things that nobody would ever want to call a "surface." At one extreme, as you noted in your comment, you could take $r$ to be a constant map. At the other extreme, take $A = T = [0,1]\times[0,1]$, let $f\colon [0,1] \to [0,1]\times[0,1]$ be a continuous surjective map (a "space-filling curve"), and define $r\colon T\to\mathbb R^3$ by $$ r(x,y) = (f_1(x),f_2(x),y), $$ where $f(x) = (f_1(x),f_2(x))$. Then the image of $r$ is the entire unit cube $[0,1]\times[0,1]\times[0,1]$.
Your comment suggested that the text then goes on to define a "regular surface" to be something with more stringent conditions designed to rule out these kinds of odd examples. But it makes no sense to call these things "surfaces" in any sense of the word. I would be inclined not to trust this text very much.