I resolved the following exercise. I need help reviewing it and giving the correct answer (if mine isn't correct of course).
where the following equivalence relation is defined
, if or ,
Prove that is closed.
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.