Doubt - Hausdorff Space

Question

I am solving exercise 3-17 in Lee's Topological Manifolds.

I'm having problems with question b). I can't imagine the diagonal.

Can someone help me. Thanks

Answer 1

There is no need to look at the diagonal in $X\times X$: just show that any two distinct points of $X$ have disjoint open nbhds. There are three cases; I give them here with a couple of hints.

• Both points are in $(0,1)\times(0,1)$: that case is very easy, since this subspace of $X$ just has its usual topology.
• One point is in $(0,1)\times(0,1)$, and the other is either $(0,1)$ or $(1,0)$; that takes just a little work. If $(x,y)$ is the point in $(0,1)\times(0,1)$, pay particular attention to $y$.
• The points are $(0,0)$ and $(1,0)$; that one is very easy.