Proposition 17

The product of a paracompact space and a compact space is paracompact.

Let $X$ be a paracompact space, $Y$ a compact space, $\mathscr{R}$ an open covering of $X \times Y$. For each $(x, y) \in X \times Y$ there is an open neighbourhood $V(x, y)$ of $x$ in $X$ and an open neighbourhood $W(x, y)$ of $y$ in $Y$ such that $V(x, y) \times W(x, y)$ is contained in some set belonging to $\mathscr{R}$. For each $x \in X$, the sets $W(x, y)$ as $y$ runs through $Y$ form an open covering of $Y$; hence there exist a finite number of points

such that the $W(x, y_i)$ cover $Y$. Let $U(x)$ denote then each of the open sets $U(x) \times W(x, y_i)$ is contained in a set of $\mathscr{R}$. Now let $(T_i)_{i \in I}$ be a locally finite open covering of $X$ which refines the covering $(U(x))_{x \in X}$. For each $i \in I$, let $x_i$ be a point of $X$ such that $T_i \subset U(x_i)$, and let us denote by $S_{i, k}$ the sets $W(x_i, y_k)$ corresponding to this point $x_i$ ($1 \leq k \leq n(x_i)$). Clearly the sets form an open covering of $X \times Y$ which refines $\mathscr{R}$, and the proof will be complete if we show that this covering is locally finite. Let $(x, y)$ be any point of $X \times Y$; there is a neighbourhood $Q$ of $x$ which meets only a finite number of sets $T_i$, and therefore the neighbourhood $Q \times Y$ of $(x, y)$ meets only a finite number of sets $T_i \times S_{i, k}$.

On the other hand the product of two paracompact spaces need not be paracompact (see Chapter IX, , Exercise 16)