Intuition behind the definition of a surface From the book Elementary Differential Geometry by Andrew Pressley, this is the definition of a surface:

A subset $S$ of  $\mathbb{R^3}$ is a surface if, for every point $p \in S$, there is an open set $U$ in $\mathbb{R^2}$ and an open set $W$ in $\mathbb{R^3}$ containing ${P}$ such that $S \cap W$ is homeomorphic to $U$.

I am struggling to understand this definition.
Why do we consider an open set $U$ in $\mathbb{R^2}$
Why do we consider the intersection of $S$ and $W$?
What is the motivation behind this definition?
 A: What's the motivation? A surface should "look locally" like $\mathbb{R}^2$: if you zoom way in, it's "flat." That means every point should have a neighborhood that's homeomorphic to $\mathbb{R}^2$.
So why $U$? In fact we could eliminate this and just demand that $S \cap W$ be homeomorphic to $\mathbb{R}^2$. Since $\mathbb{R}^2$ has the useful property that every point has arbitrarily small neighborhoods which are themselves homeomorphic to $\mathbb{R}^2$, the two definitions are equivalent. Often the definition where we consider $U$ arbitrary in $\mathbb{R}^2$ is easier to check, but it's just a matter of taste.
Why the intersection? Well, that's just the definition of the topology on $S$. A subset of $S$ is "open in $S$" if and only if it is the intersection of an open subset of $\mathbb{R}^3$ with $S$. Note that such a set is never open in $\mathbb{R}^3$ itself.
So, in summary, the definition is just saying every point in $S$ has an open neighborhood (open in the sense of the topology of $S$) that looks like $\mathbb{R}^2$ (or the seemingly-more-general-but-in-fact-equivalent requirement "looks like $U$ for some $U$ open in $\mathbb{R}^2$).
A: This basically says that a set $S \subset \mathbb{R}^3$ is a surface if:
If you look only at the part of $S$ surrounding some point $p \in S$, that small part of $S$ looks (topologically) like flat two-dimensional space.
The role of the set $W$ is to cut out the parts of $S$ "surrounding" point $p$. Usually, $W$ will be a small open ball around $p$. $U$ is the subset of the flat two-dimensional space $\mathbb{R}^2$ to which the part of $S$ surrounding $p$ (i.e. $S \cap W$) is compared.
A: It may help to think about this definition on an example. Let's assume that $S$ is a sphere in $\mathbb{R}^3$ and let $p\in S$. You can think of $W$ as a small portion of space that contains $p$ (a small ball for example). Now $S\cap W$ looks like a small "patch" on the sphere. And we notice that this small patch is approximately flat. If it's small enough (so if $W$ is small enough), you wouldn't distinguish it from some small subset of $\mathbb{R}^2$. Actually, it is homemorphic to a small subset $U$ of $\mathbb{R}^2$. This means that $S\cap W$ can be transformed in a continuous, bijective way, into $U$. So locally, the sphere looks like a plane, topologically speaking.
And the term "locally" is important, because you can't do this for the entire sphere, you won't transform it into a plane with a continuous, bijective transformation. Yet, for every point of the sphere, there is a small neighborhood around this point where you can do it.
More generally, $S$ is a surface if it locally looks like a plane, around every point.
When you draw a circle in $\mathbb{R}^2$ and you look at some small portion of this circle, it looks exactly like a segment, which is a small portion of $\mathbb{R}^2$. If you twist it a little (homeomorphism), it'll be a segment. For surfaces it's exactly the same principle, with one more dimension.
