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March 20th, 2018, 10:01 PM   #1
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understanding quotient topology

Let $h$ be a (real-valued) 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.
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March 20th, 2018, 10:04 PM   #2
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Quotient with respect to what equivalence relation?
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March 20th, 2018, 10:05 PM   #3
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Sorry I just edited my post to the quotient topology definition I am using.
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March 21st, 2018, 04:47 AM   #4
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Can you show h is closed? Hint: the domain is compact.
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March 21st, 2018, 05:38 AM   #5
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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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