Maybe this question is a little bit naive, but there might be a good explanation. I am reading about manifolds, manifolds of dimension $d$ are defined, for example to be subsets $M \subseteq \mathbb R^n$ such that for every $p \in M$ there exists an open subset $U \subseteq \mathbb R^n$ and an open subset $V \subseteq \mathbb R^d$ such that $U \cap M$ and $V$ are homeomorphic. Then notions as charts (or coordinates) and atlases are introduced.
Often an analogy is brought up to atlases as how we know them from geography classes at school, i.e. books which chart the earth. But in these, the patches that are charted, for example a map of some portion of asia, are rectangles, and rectangles more closely resemble closed sets (or compact sets), i.e. they have a boundary and contain every limit point. So taking this analogy verbal a more suitable definition might be to define a manifold of dimension $d$ to be a subset $M \subseteq \mathbb R^n$ such that every point of $M$ is homeomorphic to a rectangle $[a_1, b_1]\times \ldots \times [a_d, b_d]$ of $\mathbb R^d$. But why is this definition not used? To summarize:
Why we require every point of a $d$-dimensional manifold to have a neighbourhood homeomorphic to an open subset of $\mathbb R^d$ and not (which is more in the spirit of real-world atlases) to require a subset around every point that is homeomorphic to a $d$-fold rectangle of $\mathbb R^d$?