Find example of a mapping that is open but not continuous and a mapping that is closed but not continuous.
I striving with these questions.
I thought of using for the first case a map between $(\mathbb{Z},\tau)$ where $\tau$ is the co-finite topology to $(\mathbb{R},\tau')$ with the standard topology. I think the mapping would not be continuous and since any open set in $\mathbb{R}$ is infinite and uncountable and the inverse image would need to be countable. However I am not seeing how to prove this with more rigorous mathematical terminology and I do not see how this mapping might be open.
Question:
Can someone help me solve this question?
Thanks in advance!