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.)


*

*$f$ 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! 

$1.$ Let $X$ be the set of real numbers, with the discrete topology, and let $Y$ be the reals, with the ordinary topology. Let $f(x)=x$. Then $f$ is continuous, since every subset of $X$ is open. But $f^{-1}$ is not continuous.
$2.$ In doing $1$, we have basically done $2$. 
