Does the supremum only apply to $\mathbb{R}$ and not to $\mathbb{R}^n$. My professor stated that the supremum only applies to $\mathbb{R}$ and not to $\mathbb{R}^n$.
This got me thinking what exactly would the supremum mean in regards to $\mathbb{R}^n$?  Is there a concept that is analogous to the supremum in $\mathbb{R}^n$?
For example, suppose that $A$ is a clopen subset of $\mathbb{R}^n$ where $\emptyset \ne A \ne \mathbb{R}^n$ and $a \in A$ and $b \in \mathbb{R}^n \backslash A$.  Let's suppose that $a < b$.
What would $c=$sup$(A \cap [a,b])$ even mean?  Is this supposed to be the largest intersection?  If so wouldn't $c=$max$(A \cap [a,b])$ make more sense?  Or is there a better notation?
Thank you.
 A: If you are willing to venture a bit into the non-standard, I would say that the supremum on $\mathbb{R}^n$ does exist, although I have not normally seen the concept used in practical maths.
First, some base facts. The word supremum is a concept defined in the context of orderings on sets. The supremum property is that "every set has a least upper bound". This is true of the extended reals, $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty,\infty\}$. (And though almost true, not quite true of the reals themselves.)
In order to ask, "does the supremum exist in $\mathbb{R}^n$ too?", we have to first have an ordering on $\mathbb{R}^n$ -- what it means for $x$ to be less than $y$. There are two such orderings that are most natural and common:
1. Coordinate-Wise Ordering This is a partial ordering on $\overline{\mathbb{R}}^n$.
Let us consider the example of $\overline{\mathbb{R}}^3$.
We define that $(x,y,z) \le (x',y',z')$ whenever $x \le x'$ AND $y \le y'$ AND $z \le z'$.
2. Lexicographic Ordering This is a total ordering on $\overline{\mathbb{R}}^n$.
The way it works is like ordering words in a dictionary.
We define that $(x,y,z) \le (x',y',z')$ as follows.
First, we compare the first coordinate. If $x < x'$, then $(x,y,z) \le (x',y',z')$ (regardless of $y,z,y',z'$).
Second, if $x = x'$, then we look to the second coordinate as a tie-breaker: if $x = x'$ and $y < y'$, then $(x,y,z) \le (x',y',z')$. And so on. For instance, $(1,4,3) \le (1,5,1)$, and $(2,10,10) \le (3,1,1)$.
This ordering includes the ordering in (1), but includes a lot more as well. This ordering also distinguishes between first coordinate, second coordinate, and so on, so in this sense it is not symmetric (it treats different dimensions differently).
Under both of these orderings, the supremum of any set exists.
To find the supremum in ordering (1), one takes the supremum in each coordinate of the points' values in that coordinate. To find the supremum in ordering (2), one takes the supremum in the first coordinate to first, and then the supremum in the second coordinate of points whose first coordinate is the first supremum, and so on.
Thus it can be meaningful to speak of the supremum in $\mathbb{R}^n$.
