Point-finite collection

Theorem — ^[3][4] A topological space ${\displaystyle X}$  is normal if and only if each point-finite open cover of ${\displaystyle X}$  has a shrinking; that is, if ${\displaystyle \{U_{i}\mid i\in I\}}$  is an open cover indexed by a set ${\displaystyle I}$ , there is an open cover ${\displaystyle \{V_{i}\mid i\in I\}}$  indexed by the same set ${\displaystyle I}$  such that ${\displaystyle {\overline {V_{i}}}\subset U_{i}}$  for each ${\displaystyle i\in I}$ .