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 March 20th, 2018, 10:01 PM #1 Newbie   Joined: Oct 2017 From: sweden Posts: 20 Thanks: 0 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. March 20th, 2018, 10:04 PM #2 Senior Member   Joined: Aug 2012 Posts: 2,193 Thanks: 645 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: 752 Thanks: 257 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. Tags quotient, topology, understanding Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post aminea95 Topology 2 January 20th, 2018 07:36 AM LordofthePenguins Topology 8 July 9th, 2013 12:44 PM vercammen Topology 1 October 19th, 2012 11:06 AM lorereds Topology 0 September 7th, 2011 06:13 AM genoatopologist Topology 0 December 6th, 2008 10:09 AM

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