Relations between the real projective space and canonical covering projections

Let be the canonical covering projection. For any map define the space

x .

and the covering projection

i) What is the fibre of ?

ii) For any space establish a one-one correspondence between maps and pairs of maps such that .

iii) Let , the space with two points . For define a homeomorphism x such that

x .

I think i) is relatively straightforward and I get for a given there are two points with such that . So .

However I'm struggling with the next two, any help appreciated...