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Mikasa
Mikasa
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I have some troubles to understanding something:

We were asked to find a function that is openopen and continuouscontinuous but not closednot closed and actually I found such a function, but our tutor gave us this example

$ e^x$ since this function is continuous as a mapping $exp: \mathbb{R} \rightarrow \mathbb{R}_{>0}$$$\exp: \mathbb{R} \rightarrow \mathbb{R}_{>0}$$ and the inverse function is continuous too, so this function is also open.

and the counterexpample was, that $\mathbb{R}$ is closed and is mapped to $\mathbb{R}_{>0}$ which is open.

then i thought, hey if the inverse function is continuous that means that the fiber of closed sets have to be closed, but $ln^{-1}(\mathbb{R})$$\ln^{-1}(\mathbb{R})$ is not closed at all. where am i wrong?

I have some troubles to understanding something:

We were asked to find a function that is open and continuous but not closed and actually I found such a function, but our tutor gave us this example

$ e^x$ since this function is continuous as a mapping $exp: \mathbb{R} \rightarrow \mathbb{R}_{>0}$ and the inverse function is continuous too, so this function is also open.

and the counterexpample was, that $\mathbb{R}$ is closed and is mapped to $\mathbb{R}_{>0}$ which is open.

then i thought, hey if the inverse function is continuous that means that the fiber of closed sets have to be closed, but $ln^{-1}(\mathbb{R})$ is not closed at all. where am i wrong?

I have some troubles to understanding something:

We were asked to find a function that is open and continuous but not closed and actually I found such a function, but our tutor gave us this example

$ e^x$ since this function is continuous as a mapping $$\exp: \mathbb{R} \rightarrow \mathbb{R}_{>0}$$ and the inverse function is continuous too, so this function is also open.

and the counterexpample was, that $\mathbb{R}$ is closed and is mapped to $\mathbb{R}_{>0}$ which is open.

then i thought, hey if the inverse function is continuous that means that the fiber of closed sets have to be closed, but $\ln^{-1}(\mathbb{R})$ is not closed at all. where am i wrong?

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user66906
user66906

Open, closed and continuous

I have some troubles to understanding something:

We were asked to find a function that is open and continuous but not closed and actually I found such a function, but our tutor gave us this example

$ e^x$ since this function is continuous as a mapping $exp: \mathbb{R} \rightarrow \mathbb{R}_{>0}$ and the inverse function is continuous too, so this function is also open.

and the counterexpample was, that $\mathbb{R}$ is closed and is mapped to $\mathbb{R}_{>0}$ which is open.

then i thought, hey if the inverse function is continuous that means that the fiber of closed sets have to be closed, but $ln^{-1}(\mathbb{R})$ is not closed at all. where am i wrong?