Let $X$ a ind-scheme of ind-finite type and ind-affine. (e.g, take a k- smooth, affine scheme of finte type $T$, $C$ a smooth projective curve over $k$ and $x$ a closed point, then $X=T(C-x)$ verifies all the properties

Let $Y\subset X$ a closed subscheme of $X$, do we know if $Y$ is locally of finite type?

`$T(C-x)$`

do you mean`$\text{Hom}(C-x,T)$`

? – Matthieu Romagny Jun 8 '13 at 9:48