I solved the following exercise (but i am not sure for the solution).
Can you give me a hand?Thank you.
Given the square
Let where we define the following equivalence relation:
a) Show that is isomorphic to .
b) Prove that is closed.
c) Prove that is not open.
a) first of all we construct the two following applications
and where .
Then we construct pasting together .Since and satisfy the conditions of The Pasting Lemma will be continue. In addition it is bijective and the equivalence relation that induces coincides with the equivalence relation defined on .Then is isomorphic to .
b) is connected and compact, so also will be connected and compact. We must show at this point that is (Hausdorff). That is to prove that the set
is a closed .
where are the vertices of the square.
is closed because is . , are closed because they are closed subsets of a space and images of the compact of through continuous applications. is closed because it is a finite set of points in space . Since , then also will be closed, this means that is . We conclude that is closed.
c) Then we know that if is an open set in (in our case) is said to be an open application if will be open. Then we take where as a general case assuming that none of the parties are empty. Similarly . At this point we compute and then and since and are closed the application can not be open. In fact, the application will be closed (if is closed).