Hat notation on $\mathbb{R}^n$ vectors

Question

I've just read the following passage from an article:

... then there exist and interval $I \subseteq \mathbb{R} $, an open set $A \subseteq \mathbb{R}^{n-1}$ and a function $f: A \rightarrow I$ such that:

• ... $\{x; x_i \in I, (x_1,....,\hat{x_i},...,x_n) \in A \}$ ...
• ... $\{x; x_i=f(x_1,....,\hat{x_i},...,x_n), (x_1,....,\hat{x_i},...,x_n) \in A\}$ ...

What does the hat notation means?

Answer 1

Omit the term with hat. For example, $(x_1,x_2,\hat {x_3},x_4)$ would mean $(x_1,x_2,x_4)$.

Answer 2

The hat signifies that the corresponding coordinate is omitted. Note that $x \in \mathbb R^{n}$ and $A \subseteq \mathbb R^{n-1}$ in the definition of $\Omega_0$. By omitting one coordinate $x$ becomes a vector in $\mathbb R^{n-1}$.

Answer 3

It sometimes denotes unit vectors, although here, I would say it’s different, since only the component has a hat. In the context, I would say it means that you remove the $i$-th component from the vector, since $A$ is a subset of $\mathbb{R}^{n-1}$.