Let $H=[-1,1]\times \{0\}$ and $V=\{0\}\times [-1,0)$ in the plane. Let $T=H \cup V$. Show that $T$ is not homeomorphic to the unit interval $I=[0,1]$.
My idea for this problem is that , if we remove a point from the unit interval , we will be left with at most two connected components, but if we remove the origin from $T$ we will be left with $3$ connected components. Is this enough to prove that $I$ and $T$ are not homeomorphic ? How should I write my answer rigirously?
Any help is appreciated, Thanks !
Best Answer
Yes, your proof is correct, but from your comment above, it seems that the reason why it is correct is not completely clear to you. What you are implicitly using is the following proposition.
Note: The conditions we need to check are about the spaces $X\setminus\{x\}$ and $Y\setminus\{y\}$, but our conclusion pertains to $X$ and $Y$.
I have included the proof below, you just need to move your mouse over the second grey box to see it, but I recommend you try to prove it yourself first. If you want a hint, move your mouse over the first grey box.