I am looking for a precise definition of a rectangular neighborhood in the definition of a foliation chart.
In the book "Foliations I" by Alberto Candel and Lawrence Conlon, the definition is the following(same as in the English Wikipedia):
"A rectangular neighborhood in $\mathbb{F}^n$ is an open subset of the form $B=J_1\times\dotsc \times J_n$, where each $J_i$ is a (possibly unbounded) relatively open interval in the $i$th coordinate axis."
I have no idea what relatively open should mean here.
Usually, that means to be open in a closed subspace, but it could also mean that one allows precisely all non-closed intervals (This makes a difference as one must be connected and the other one not).
The definition goes further as follows:
"If $J_1$ is of the form $(a,0]$ we say that $B$ has boundary $\partial B=\{(0,x^2,\dotsc,x^n)\in B\}$."
Well, this is impossible, because if $J_1$ would have this form, then $B$ would not be an open subset of $\mathbb{R}^n$.
Furthermore, the definition tells us, that a boundary does only exist if $J_1$ is of this form, so a boundary doesn't exist for example in the case $J_2=(0,b]$, right!?