Take a point $x\in X_{\alpha}$ and a point $y$ or a closed set $A$ contained in $X_{\alpha}-\{x\}$ (if $X_{\alpha}$ contains one point only, we are done). For every $\beta\neq\alpha$ fix $x_{\beta}\in X_{\beta}$ . For every $z\in X_{\alpha}$ let $p_{z}$ be the product of $z$ with $x_{\beta}$ for all $\beta\neq\alpha$ . For a subset $S\subseteq X_{\alpha}$ let $P_{S}=S\times\sqcap_{\beta\neq\alpha}X_{\beta}$ . If the product is Hausdorff, $p_{x}$ and $p_{y}$ can be separated by basis neighborhoods $U$ and $V$ . Then $\pi_{\alpha}(U)$ and $\pi_{\alpha}(V)$ are open (the projection is an open map) and disjoint (if there is a common point $z$ then $p_{z}$ belongs to both $U$ and $V$ ) neighborhoods of $x$ and $y$ . Now, if the product is regular, then there are disjoint basis neighborhoods of $p_{x}$ and $P_{A}$ , and their projections into $X_{\alpha}$ are open and disjoint (for the same reason) neighborhoods of $x$ and $A$ . If the product is normal and $B$ is a closed subset of $X_{\alpha}$ that does not intersect $A$ , then $P_{A}$ and $P_{B}$ can be separated by disjoint basis neighborhoods whose projections into $X_{\alpha}$ separate $A$ and $B$ .