March 20th, 2018, 10:01 PM  #1 
Newbie Joined: Oct 2017 From: sweden Posts: 20 Thanks: 0  understanding quotient topology
Let $h$ be a (realvalued) continuous function on some closed interval $[c,d].$ Let, $\lambda =\min_{[c,d]}h$ and $\gamma = \max_{[c,d]}h$ . Also, $h$ can be considered as a surjective map (onto) from $[c,d] \rightarrow [\lambda,\gamma]$. **Question** How can one show that if $[c,d]$ has the usual topology, then the quotient topology on $[\lambda,\gamma]$ is also the usual topology ? I am genuinely frustrated because this is an example from the quotient topology chapter, but the solution to this the author gave I cannot understand at all. I would really appreciate some help. Note the quotient topology I am working with here is given by: a quotient topology on $Y$ is defined to be $T_Y=\{ V\subset Y : f^{1}(V) \in T_X\}$,where $f:X\rightarrow Y$, and a topological space $X$ with topology $T_X.$ Last edited by heinsbergrelatz; March 20th, 2018 at 10:04 PM. 
March 20th, 2018, 10:04 PM  #2 
Senior Member Joined: Aug 2012 Posts: 1,960 Thanks: 547 
Quotient with respect to what equivalence relation?

March 20th, 2018, 10:05 PM  #3 
Newbie Joined: Oct 2017 From: sweden Posts: 20 Thanks: 0 
Sorry I just edited my post to the quotient topology definition I am using.

March 21st, 2018, 04:47 AM  #4 
Senior Member Joined: Oct 2009 Posts: 430 Thanks: 144 
Can you show h is closed? Hint: the domain is compact.

March 21st, 2018, 05:38 AM  #5 
Newbie Joined: Oct 2017 From: sweden Posts: 20 Thanks: 0 
I am sorry my topplogy is very weak. My midterm is 2 days from now, and this is one of the past problems. Can you show me how to do this, I really have to understand how in a short time and the solution manual is not helping one bit.


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quotient, topology, understanding 
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