# Thread: Product of Hausdorff spaces

1. ## Product of Hausdorff spaces

I have to prove that if $X_\alpha$ is Hausdorff for each $\alpha \in A$, then $X= \Pi_{\alpha \in A}X_\alpha$ is also Hausdorff.

I have proved this for the two dimensional case, and I know if A is finite then it would be true for all of them by induction, but it doesn't say anything about A being finite or infinite. Is this still true by induction if A is infinite?

2. No. Induction on n shows that "P(n)" is true for all finite n- it does not extend to infinite values.

3. Take a pair of distinct points $x, y \in X$. Then for some index $\beta \in A, x_\beta \neq y_\beta$, where $x_\beta, y_\beta \in X_\beta$. Since $X_\beta$ is Hausdorff, we can find disjoint open subsets $U$ and $V$ of $X_\beta$ containing $x_\beta$ and $y_\beta$ respectively.

Now let $U_\alpha = X_\alpha$ for $\alpha \neq \beta$, and $U_\beta = U$, and similarly take $V_\alpha = X_\alpha$ for $\alpha \neq \beta$, and $V_\beta = V$. Then $\Pi_{\alpha \in A}U_\alpha$ and $\Pi_{\alpha \in A}V_\alpha$ are disjoint open subsets of $X$ containing $x$ and $y$ respectively, showing that $X$ is Hausdorff.