About a counterexample for the Open Mapping Theorem

Question

I want to ask some thing about a counterexample for the open mapping theorem:

Find a discontinuous linear mapping $T: \ X \to Y$ such that $T(X)=Y$ and $X,Y$ are Banach but $T$ is not open.

I find there is a answer in the link below:

Counterexample for the Open Mapping Theorem

But I don't understand how to prove clearly three cases b), c) and d), especially the case c).

Anyone can help me?

Answer 1

If $T : X\rightarrow Y$ is linear bijection between Banach spaces $X$ and $Y$, then the following are equivalent

1. $T$ is continuous.
   
2. $T^{-1}$ is continuous.
   
3. $T$ is an open map.
   
4. $T^{-1}$ is an open map.
   
5. $T$ has a closed graph.
   
6. $T^{-1}$ has a closed graph.

You may want to read the previous post in light of this.