# 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?

Your tutor is wrong. Every homeomorphism (such as $\exp:\mathbb R\to\mathbb R_{>0}$) is closed. The set $\mathbb R_{>0}$ is a closed set in the standard topology on $\mathbb R_{>0}$.

• okay, thank you. this really confused me! – user66906 May 17 '13 at 9:30

If you consider $\def\R{\mathbb{R}}\def\exp{\mathrm{exp}}\exp\colon\R\to\R$, then the example is correct: the map is not closed, because

$$\exp(\R)=\R_{>0}$$

which is not closed in $\R$. On the other hand, the image of an open set in $\R$ is open in $\R_{>0}$, so also open in $\R$. Hence the function is open but not closed.

If you consider $\exp\colon\R\to\R_{>0}$, then of course its image is both open and closed in the codomain, being the whole of it. As already observed, this is a homeomorphism, so it's obviously open and closed.

I think you can consider the following function instaed: $$\pi:\mathbb R^2\to\mathbb R\\\\ \pi\left((x,y)\right)= x$$

• Nice! I miss you! ;-) – Namaste May 18 '13 at 1:27
• Do not fret! You're never "too late" in my book ;-) How are you? Did you sleep well? – Namaste May 18 '13 at 4:11
• Oh my! Due you have a "due date" when you need to complete your paper? – Namaste May 18 '13 at 4:20
• Oh, I understand. Mathematics can be very strict to those who choose to serve. I will certainly wish you luck, and share with you my faith in your resilience to conquer the task! And just imagine the joy you will feel when you do so!! – Namaste May 18 '13 at 4:30
• And be just as kind to yourself as you are to others! ;-) – Namaste May 18 '13 at 4:36