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How to show that $(0,1)$ and $[0,1)$ are not homeomorphic?

My attempt:

On contrary suppose it is then

there exist $f:[0,1)\to (0,1)$ which is continuous

But inverse image of open set is open

So contradiction

Is my argument is correct?

Curious student
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    Your argument is not correct. Are you familiar with connectivity? – drhab Dec 04 '18 at 12:14
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    No Sir.Is that connectedness – Curious student Dec 04 '18 at 12:16
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    I think so (English is not my language). If $0$ is removed from $[0,1)$ then what remains is $(0,1)$ which is connected. But it is not possible to remove a single point from $(0,1)$ such that the remaining space is connected. – drhab Dec 04 '18 at 12:19

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