Proposition 2

Every regular Lindelöf space is paracompact.

Soit $X$ a regular Lindelöf space, and $\mathscr{R}$ an open covering of $X$. For every $x \in X$, let $U_x \in \mathscr{R}$ containing $X$, and let $V_x$ be an open neighbourhood of $x$ such that $V_x \subset U_x$. Since $X$ is a Lindelöf space, there exists a sequence $(x_n)_{n \geq 0}$ of points of $X$ such that the $V_{x_n}$ form a covering of $X$. Put then, for every integer $n \geq 0$, $W_n = U_{x_n} \cap \complement \left( \bigcup_{k < n} V_{x_k} \right)$. The sets $W_n$ are open and form a covering of $X$ finer than $\mathscr{R}$. On the other hand, this covering is locally finite, since every $x \in X$ belongs to some $V_{x_i}$ for at least one $i$; then $V_{x_i}$ is a neighbourhood of $x$, which meets no set $W_n$ for $n > i$.