Show that a non -empty algebraic variety can be uniquely written as a finite union of irreducible closed sets which do not contain each other.

I have a question in the proof ofCorollory $4.4$ of Daniel Perrin's Algebraic Geometry on Page $45$.

Corollary $4.4$ A non empty algebraic variety can be uniquely written as a finite union of irreducible closed sets which do not contain each other.

Proof:By quasi-compactness, we can write $X= U_1\cup ...U_n$ where the sets $U_i$ are open affine sets.We can write each of the sets $U-I$ as finite union of closed sets $U_{i,j}$ which are closed in $U_i$ (How?)

Can you please tell how sets $U_i$ can be written as finite union of closed sets $U_{i,j}$ which are closed in $U_i$ ? Which result should I use?