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What is an interval in $\mathbb{R}^n$?

Isn't interval just in $\mathbb{R}$ and higher dimensions have some sort of rectangle or cube?

Particularly here it's said that:

Let $I$ be a bounded closed interval in $\mathbb{R}^n$ . Then $μ( \partial I ) = 0$.

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    It seems that in the paper you linked, the word *interval* is used for [*n-cell*](https://en.wikipedia.org/wiki/K-cell_(mathematics)). – 2017-01-26
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    Have you taken a look at [wiki](https://en.wikipedia.org/wiki/Interval_(mathematics)#Multi-dimensional_intervals) ? They define an interval in $\mathbb{R}^n$ as a cartesian product of intervals in $\mathbb{R}$. All of the types of intervals of real numbers they give are contained in this definition of an interval in $\mathbb{R}$ : $I$ is an interval if and only if for all $a,b,c$ such that $a,c\in I$, $a$b\in I$. – 2017-01-26
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    A certain type of parallipiped (or parallelepiped). – 2017-01-27

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An interval $I$ in ${\mathbb{R}}^n$ is a set $$ I = I_1 \times \ldots \times I_n\mbox{,} $$ where $I_i$ is an interval in $\mathbb{R}$, $i = 1 , \ldots , n$, being "$\times$" the cartesian product for sets.

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    But $I_1 \times I_2$ is a rectangle for example. So why call it an interval? – 2017-01-26
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    mavavilj I guess it's because just saying "interval" is easier than worrying about using the right word for the n-dimensional object, like saying Cube instead of rectangle etc – 2017-01-27
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    Normally, mathematicians try to construct a theory that is common in all dimensions and therefore has the same name, but in particular different names are used when studying cases like $ n = 1, 2, 3 $, because they are the most interesting cases to make practice problems (calculus for derivates, integrals,...). For example, in differential geometry, a curve is not the same as a surface, but both are manifolds. – 2017-01-27