One point compactification of $[0,1]\times [0,1)$

Question

What is one point compactification of $[0,1]\times [0,1)$?

If we draw the figure we see that top line is missing and we've to add just one point to make it compact. So I think triangle will be the answer but I'm not sure.

Answer 1

You are correct. To prove it, suppose that $K\subseteq[0,1]\times[0,1)$ is compact, and show that there is a $y\in[0,1)$ such that $K\subseteq[0,1]\times[0,y]$. This will allow you to construct a simple homeomorphism between the one-point compactification and the closed triangular region

$$\{\langle x,y\rangle\in\Bbb R^2:x,y\ge 0\text{ and }x+y\le 1\}\;.$$