Exercise with homeomorphism, open and closed: topology

Hello,

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:

where or

a) Show that is isomorphic to .

b) Prove that is closed.

c) Prove that is not open.

__ Solution __:

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).

Re: Exercise with homeomorphism, open and closed: topology

Quote:

Originally Posted by

**mameas** 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

.

This looks good to me.

Quote:

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.

Ok, so you're using the theorem that if is Hausdorff, compact and continuous then is closed, I assume? But then, I don't understand the problem. You just proved that and since Hausdorffness is preserved under homeomorphism (in fact, it is preserved under open surjections with closed kernels) you know that is Hasudorff, no?

Quote:

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).

Good, assuming your readership will understand that you are using the fact that since is connected there exists no nontrivial clopen sets.

Good job overall!

Re: Exercise with homeomorphism, open and closed: topology

Perhaps, following the rules of the forum I do not respond to my thread before 24 hours, however I want to thank you Drexel28.

About the question b) of the exercise I agree totally with you, but I prefer to answer the differents questions of the exercises independently (when it is possible of course). Thank you again.

Re: Exercise with homeomorphism, open and closed: topology

Quote:

Originally Posted by

**mameas** Perhaps, following the rules of the forum I do not respond to my thread before 24 hours, however I want to thank you Drexel28.

About the question b) of the exercise I agree totally with you, but I prefer to answer the differents questions of the exercises independently (when it is possible of course). Thank you again.

No haha, you can respond as soon as you like! And if you want to thank a user merely press the thank you button in the bottom left corner of their post.