e $Y$ is a closed discrete subspace of $X$ of size $\aleph_1$. Then there is a locally compact normal space $X'$ with a closed discrete subspace $Y'$ of size $\aleph_1$, such that if $Y'$ is separated in $X'$, then $Y$ is separated in $X$, but each point in $Y'$ has character $\leq\aleph_1$.

By Watson’s character reduction technique [@W], there is a discrete collection of compact subsets of $X$, $\mathcal{K}=\{K_y:y\in Y\}$, such that $y\in K_y$, and each $K_y$ has character $\leq\aleph_1$. Let $X'$ be the quotient of $X$ obtained by collapsing each $K_y$ to a point $y'$. This collapse is a perfect map, so preserves normality and local compactness, and it is clear that $\{y':y'\in Y\}$ is separated if and only if $\{K_y:y\in Y\}$ is separated, and that $Y$ is separated if $\{K_y:y\in Y\}$ is.

\[lem47\] Suppose $X$ is a locally compact normal space of Lindelöf degree $\aleph_1$ with an uncountable closed discrete subspace. Then there is a continuous image of $X$ of weight $\aleph_1$ enjoying the same properties.

Let $\mathcal{U}$ be an open cover of $X$ of size $\aleph_1$ with each member of $\mathcal{U}$ a cozero set with compact closure. Without loss of generality, assume that for each $x\in X$ there is a $U\in\mathcal{U}$ such that $x\in U$ and $U$ meets at most one element of a given closed discrete set $D$ of size $\aleph_1$. Also without loss of generality, assume $\mathcal{U}$ is closed under finite intersections. For each $U\in\mathcal{U}$, let $f_U:X\to[0,1]$ with $U=f^{-1}_U((0,1])$. Define an equivalence relation on $X$ by letting $x_0\!\!\sim\!\! x_1$ if $f_U(x_0)=f_U(x_1)$ for all $U\in\mathcal{U}$. Let $X/\!\!\sim$ be the quotient set, with $\pi:X\to X/\!\!\sim$ the projection. Topologize $X/\!\!\sim$ by taking as base all sets of form $\pi(U)$, $U\in\mathcal{U}$. Then $X/\!\!\sim$ is $\textup{T}_{3 \frac{1}{2}}$ and of weight $\leq\aleph_1$. To see the former, consider $X$ as embedded in $[0,1]^{C^*(X)}$ by $e(x)=(f(x))_{f\in C^*(X)}$. Let $p:[0,1]^{C^*(X)}\to[0,1]^{\{f_U:U\in\mathcal{U}\}}$ b