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 and 2 are really the same thing. Also, what do you mean by the inverse of a function if it is not a bijection (your question 3)? The inverse of a function is not defined for something that is not bijective.
For an example of 1 take $X = \mathbb{Q}$ with the discrete topology and $Y = \mathbb{Q}$ with the usual topology inherited from $\mathbb{R}$. Let $Id: X \rightarrow Y$ be the identity map on the level of sets. This map is clearly continuous and a bijection. But, it is not open hence the inverse is not continuous.
