Counter-examples of homeomorphism Briefly speaking, we know that a map $f$ between $2$ topological spaces is homeomorphic if $f$ is a bijection and the inverse of $f$ and itself are both continuous.
So, can anyone give me $2$ counter examples(preferably simple ones) of non-homeomorphic maps $f$ between 2 topological spaces that satisfy the properties I give? (Only one of them is satisfied and 3 examples for each property.)


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*$f$ is bijective and continuous, but its inverse is not continuous.

*$f$ is bijective and the inverse is continuous, but $f$ itself is not continuous.


In addition, can we think about some examples of topologies that are path-connected?
I will understand the concept of homeomorphism much better if I know some simple counterexamples. I hope you can help me out. Thanks! 

$f:[0,2\pi)\to S^1$ given by $f(t)=(\cos(t),\sin(t))$ is continuous and bijective but its inverse is not continuous. 
The inverse of $f$ is an example of a bijection $f^{-1}:S^1\to [0,2\pi)$ that is not continuous, but its inverse is continuous. 
It is very illustrative to think geometrically about this example. It just involves gluing the interval $[0,2\pi)$ to get the circle.
If $f$ has an inverse then $f$ is a bijection. 
