Is an open interval homeomorphic to a closed one?

Question: Are $[a,b]$ and $(c,d)$ homeomorphic?

Solution: No. A quick way to see that is to observe that $[a,b]$ is compact and $(c,d)$ isn't. Independent from the buildup necessary for this, one might consider where $a$ goes under this map and what happens to its neighborhoods.

• What about a homeomorphism between $[a,b]$ and $[c,d)$?