The set of all $k$-dimensional planes which intersects $X$ is closed in $G(k,n)$

Question

Let $X$ be irreducible algebraic set of projective n space. I am trying to show that: The set of all $k$-dimensional planes which intersects $X$ is closed in $G(k,n)$, where $G(k,n)$ is the Grassmanian of the linear subspaces of dimension $k$ inside projective $n$ space. I know the fact that $A=${$(H,x)$|$x$ belongs to $H$} is closed in the direct product of $G(k,n) $ and the projective $n$ space, where $H$ belongs to $G(k,n)$. Using the above known fact I am trying to show that : The set of all $k$-dimensional planes which intersects $X$ is closed in $G(k,n)$

Answer 1

The projection map from direct product of $G(k,n)$ and the projective $n$ space to $G(k,n)$ is a closed map since projective $n$ space is compact. Hence the result follows easily