Notation for the closed set $[a,b]$ and set $(a,b)$ in $\mathbb R^2$

In one dimensional space, $(a,b)$ and $[a,b]$ are common.

Now consider $a,b\in\mathbb R^2$,

Define $[a,b]$ be the segment $\overline{ab}$. Let $(a,b)$ be the segment without the endpoints.

Are these notations standard? Has these notations be used before?

One thing worried me is that $(a,b)$ usually means an open set and in my definition $(a,b)$ is not open. So this notation could be informal and sloppy. $[a,b]$ seems to be a closed set, though.

Solution 1:

It depends on who’s using them, but it is a typical notation that is frequently found in topics where connecting lines are important, such as convex analysis. But then there is no real mathematical standard notation for these things, as the use depends on how often one needs such things. In linear algebra $[a,b]$ could also mean the vector space created by $a$ and $b$ for examble.