Are all subspaces of a paracompact space normal?
This is what I think about this question... First a paracompact Hausdorff space turns out to be Normal, second the paracompact property is not hereditary, meaning any subspace need to be closed+Hausdorff to be normal. So I think the answer is no, but I have to find a counter-example.
Am I right about this and what counter-example do we have?
Every compact space is paracompact. Let $X$ be the one-point compactification of a countably infinite discrete space $X_0$, and let $Y$ be the one-point compactification of an uncountable discrete space $Y_0$. Let $p$ be the non-isolated point of $X$ and $q$ the non-isolated point of $Y$. Then $X\times Y$ is compact, so it’s paracompact. However, the subspace
$$Z=(X\times Y)\setminus\{\langle p,q\rangle\}$$
is not even normal, so it’s certainly not paracompact. Specifically, the closed sets $H=X_0\times\{q\}$ and $K=\{p\}\times Y_0$ cannot be separated by disjoint open sets.
To see this, suppose that $U$ is an open nbhd of $H$. For each $x\in X_0$ there is a finite $F_x\subseteq Y_0$ such that $\langle x,q\rangle\in\{x\}\times(Y\setminus F_x)\subseteq U$. Let $F=\bigcup_{x\in X_0}F_x$; then $F$ is countable, so there is a $y\in Y_0\setminus F$. But then $X\times\{y\}\subseteq U$, so no open nbhd of $\langle p,y\rangle$ is disjoint from $U$, and therefore no open nbhd of $K$ is disjoint from $U$.