Technical name of a specific subset

Question

Consider a set $A$ in $\mathbb{R}^3$. Let $(x,y,z)$ denote a representative element of $\mathbb{R}^3$. Let $$ A(x=3)\equiv\{(x,y,z)\in A: x=3\}. $$ Does this set has a specific technical name? For instance, something related to the notion of projection?

Answer 1

That is not a projection. A projection would be something like

$$\{(3,y,z) \mid(x,y,z)\in A\}$$

This takes all the points in $A$ and squishes them down (projects them) to have an $x$ value of $3$.

As for your set, although it's not a term I see defined "officially", I would know exactly what you meant if you were to call your set "a slice of $A$, (sliced by the plane $x=3$)".

Also, as @achillehui noted in a comment, "cross section" is another term that would make sense.