# $[0,1]$ is commonly called the unit interval - is there a similar term for $[-1,1]$?

by Candid Moon _Max_   Last Updated July 17, 2019 12:20 PM - source

The interval $$[0, 1]$$ is commonly called the 'unit interval'. Is there something similar for $$[-1, 1]$$? Like a pre-defined name.

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I don't think so. Personally speaking, I have never found a particular definition for intervals of the form $$[a,b]$$ in general. I believe that the reason of that is that any closed interval is homeomorphic to $$[0,1]$$ and then it shares the same topological properties of $$[0,1]$$.

Any pair of closed intervals $$[a,b]$$ and $$[c,d]$$ are homeomorphic, for any choice of $$a,b,c,d\in\Bbb R$$. In fact the function $$f:[a,b]\to [c,d]$$ defined as $$f(x)=\frac{x-a}{b-a}(d-c)+c$$ is a possible homeomorphism.

By taking $$c=0$$ and $$d=1$$, you find an explicit homeomorphism between $$[a,b]$$ and $$[0,1]$$.

Also, the unit interval appears in more contexts than a general closed interval. For instance in algebraic topology an homotopy is parametrised by $$[0,1]$$. Another example comes from probability theory. The probability in measure by a real number in the unit interval $$[0,1]$$. So it makes sense that $$[0,1]$$ has a privileged role and has an own definition.

InsideOut
July 17, 2019 12:00 PM

It's the closed unit ball of $$\mathbb{R}$$ in its usual absolute value norm.

In any normed space $$(X, \|\cdot\|)$$, we have a closed unit ball (or disk) $$D=\{x \in X: \|x\| \le 1\}$$ and a unit sphere

$$S=\{x \in X: \|x\|=1\}$$ The unit sphere in the reals is $$\{-1,1\}$$ of course, the boundary of the unit ball $$[-1,1]$$.

They often get indexed by their dimension $$n$$ if it is finite, so $$D^1$$ and $$S^0$$ in the above case. $$\partial D^n = S^{n-1}$$ for all $$n\ge 1$$.

Henno Brandsma
July 17, 2019 12:13 PM

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