The set of all projection maps

Question

I wanted to know if the set of all projection maps is closed, open, bounded or connected ? I see that it's closed because it's equal the reciprocal image of {0} by the continuous transformation p^2-p where p is a projection. I observe also that it cannot be connected because the application that associates a projection p with its rank=trace is continuous and has values in a discrete set {0,1....n}. How about it being open or bounded ? Thank you.

Answer 1

I presume you mean in the space of $n \times n$ matrices, $M_n(\mathbb{R}) \cong (\mathbb{R})^{n^2}$.

It cannot be open because an open and closed set in $M_n(\mathbb{R})$ must be empty or the whole space. (A consequence of connectedness of Euclidean space.)

To see that it is bounded, try producing a projection on to the first $k$ coordinates with arbitrarily large norm. For this it is convenient to express as a block diagonal matrix, where the upper left is the identity matrix,the bottom blocks are 0, and the upper right is free.