interval has one element {how to write it} symbol

Question

If we have interval has one element for example $[a,a]$ Is it wrong to write it in the form $ [a,a] $? We write it as $\{a\}$

Thank you :)

Answer 1

It is not wrong. I wouldn't use it as the go-to notation for a singleton (for which I would strongly recommend $\{a\}$), but if you are in a context where you're mainly dealing with intervals of the form $[a,b]$ for $a<b$ and you want to extend it to $a\le b$, I think no one can argue with it. The definition $$[a,b]=\{x\in\Bbb R\,:\, a\le x\le b\}$$ needs not be modified.

Added: On a side note, I would refrain from making a big deal out of the identities $$\forall a>b,\ (a,a)=(a,a]=(a,b)=[a,b]=\emptyset$$

Answer 2

The notations are equivalent. $[a,b] = \{x \in R \ | \ a \leq x \leq b\}$ so $[a,a] = \{ a \}.$

Answer 3

I think this question is more complex that what it seems. The notation $[a,a]$ would be correct if the context make clear that $[a,a]$ denote a closed interval in $\Bbb R$.

The problem is that this notation is not used so it can be very confuse.

Mathematical language is not very different that any other human language so at the end it relies in context and costume. Then if you try to communicate something the more efficient way is following the most standardized notation.

In this case the standard way to write it is $\{a\}$.