Hi, everyone,

I resolved the following exercise. I need help reviewing it and giving the correct answer (if mine isn't correct of course).

Let's have:

where the following equivalence relation is defined

, if or ,

Prove that is closed.

Solution:

is connected and compact so even must be connected and compacted. At this point we have to prove that is (Hausdorff). That is to prove that the set

is closed on .

,

.

is closed because is . , are closed being subspaces of space and immages of the compact on through continuous applications. Because then even will be closed, this means that is . I conclude that is closed.