Homeomorphism between two sets

Question

I am to prove that there is no homeomorphism between $(a,b)$ and $[a,b)$

It is defined that function is bijective, continuous, and inverse continuous.

How can I derive a contradiction assuming that there exists a homeomorphism between two sets?

One approach that I take is if f is continuous map $[a,b)$ to $(a,b)$, then $f^{-1}((a,b))$ must be open set, but $[a,b)$ is not open.

I think this way is more of like set theory rather than using definition of continuity and I doubt this completes proof or not.

Answer 1

This is not the proof. Indeed I claim that $[a,b)$ is actually open (for the induced topology on $[a,b)$ by $\mathbb{R}$).

Hint 1:

Use the fact that a continuous map sends connected sets to connected sets.

Hint 2:

Can you take one element out of $[a,b)$ such that it still connected? Same question with $(a,b)$.