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...