Proposition 3

Given any locally finite open covering $(A_i)_{i \in I}$ of a normal space $X$, there exists a continuous partition of unity $(f_i)_{i \in I}$ on $X$, subordinate to the covering $(A_i)_{i \in I}$.

By Theorem 3 there exists an open covering $(B_i)_{i \in I}$ of $X$ such that $\overline{B}_i \subset A_i$ for each $i \in I$, and it is clear that the covering $(B_i)$ is locally finite. By axiom $(O''_V)$, for each $i \in I$ there exists an open set $C_i$ such that $\overline{B}_i \subset C_i \subset \overline{C}_i \subset A_i$. By axiom $(O_V)$, for each $i \in I$ there exists a continuous mapping $g_i$ of $X$ into $[0,1]$, such that $g_i(x) = 1$ on $\overline{B}_i$ and such that the support of $\overline{g}_i$ is contained in $\overline{C}_i$, and therefore contained in $A_i$. Since $(B_i)$ is a covering of $X$, we have $\sum_{i \in I} g_i(x) > 0$ for each $x \in X$; if we put

for all $i \in I$ and all $x \in X$, then the $f_i$ form a continuous partition of unity subordinate to the covering $(A_i)$.