In our lecture notes we have the following remark:
Let $U=\{u_1,…,u_k\}$ be a finite open family and $F=\{F_1,…,F_k\}$ be a closed reduction of $U$.
Then $H=\{H_1,…,H_k\}$ such that $H_i=u_i$ or $H_i=F_i^c$ is a finite open cover of $X,∀i=1,…,k$.
How is this a cover of $X$? What does $H_i=u_i$ or $H_i=F_i^c$ mean?