In this section we will present another way of defining a trapped surface. On the figure 
one can see that the trapped surface is inside the compact set S and its domain of dependence .
One can consider the idea of cutting off a compact set S from the spacetime(by this we mean ignoring what is inside, not really cut off.). If we take the metric to be de Sitter outside of that compact set we can prove the Topological Censorship.
Figure: Representation of the compact set S around the trapped surface 
First let us show one of the lemma (the proof is givent in the appendix ).
This can be seen as the similar of the lemma two of the original paper. Our version of the Topological Censorship theorem is then:
