the set of all matrices which represent orthogonal projection in $M_n(C)$ is closed The set of all matrices which represent orthogonal projection in $M_n(C)$ is closed. I can not identify the set. Can you please help me to do so?

The set you are looking for is the following:
$$\mathrm{Gr}_n(\mathbb{C}):=\{A\in\mathcal{M}_n(\mathbb{C})\textrm{ s.t. } A^2=A,{}^\intercal\overline{A}=A\}.$$
It is closed, since it is given as an intersection of closed sets. Indeed, the two following maps $A\mapsto A^2-A$ and $A\mapsto{}^\intercal{\overline{A}}-A$ are continuous (why?).
Remark. Being a projector is exactly asking $A^2=A$ and orthogonality reads as ${}^\intercal\overline{A}A=I_n$.

If you are wondering why I called this set $\textrm{Gr}_n(\mathbb{C})$ it is because one has:
$$\textrm{Gr}_n(\mathbb{C})=\coprod_{k=0}^n\textrm{Gr}_{n,k}(\mathbb{C}),$$
where $\textrm{Gr}_{n,k}(\mathbb{C})$ is the collection of $k$-planes in $\mathbb{C}^n$.
