TITLE: An Open Interval and a Half-open Interval are not Homeomorphic
QUESTION [5 upvotes]: As the topic, how can I prove it?

Prove that (a,b) and [a,b) in R are not homeomorphic metric space.

I'm pretty weak on this section (homeomorphism) and I don't have any clue to figure it out.
Please help me, thank you.

REPLY [11 votes]: Thank to all above, now I write it this way and it seems complete.

Suppose not, i.e. if there exist $f:(a,b) \to [a,b)$ a homeomorphism.
By definition, f is continuous, $f^{-1}$ is continuous, and $f$ is bijective.
$f$ is bijective then we have the inverse image of $[a,b)$ is $(a,b)$
$f^{-1}(a)$ is a single point, say c, and $a \lt c \lt b$.
Consider $f^{-1}((a,b))=(a,c) \cup (c,b)$ is disconnected.
It contradicts the fact that $f^{-1}((a,b))$ is connected, noted that $(a,b)$ connected and $f^{-1}$ is continuous.
Therefore, $(a,b)$ and $[a,b)$ are not homeomorphic.