Hausdorffness of product space

Question

I am having trouble understanding the product space.

How do I prove that $X\times X$ Hausdorff implies that $X$ Hausdorff?

I know that for $(x,y)(x',y')$ different there exist different disjunct open subsets of $X\times X$ that contain those points. How do I get the same for just $X$?

Answer 1

Hint: Subspace of Hausdorff space is Hausdorff.

Answer 2

If $x,y \in X$, consider the pairs $(x,x)$ and $(y,y)$ in the product space. We can apply the Hausdorff hypothesis to find two disjoint open sets $W_{x}$ and $W_{y}$. Now, because of the definition of the product topology, we can find an open set $U_{x}$ of $X$ such that $(x,x) \in U_{x}\times U_{x} \subseteq W_{x}$. Do the same with $W_{y}$ and the Haussdorf hypothesis is valid by taking the open disjoint sets $U_{x}$ and $U_{y}$.

Answer 3

Hint: If $f\colon X\to Y$ is a continuous map, $x\mapsto (x,f(x))$ is a homeomorphic embedding of $X$ into $X\times Y$.