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As usual, i'm having issues understanding some basic examples D:

e: [0,1) --> S^1 is not a quotient map. Any neighborhood on the unit circle starting at 1 and going around to e^(i*2pi*c) l be a not open neighborhood (it's complement is not closed) of the unit circle who's preimage is [0,c), which is open in [0,1).

Cool! I believe that this map is not a quotient map. My book goes on to say that:

e: [0,1] -->S^1 is a quotient map because it is also a closed map. Cool, that makes sense to me! I mean before we had the closed neighborhood [a,1) that would map to the not closed neighborhood [e^(i*2pi*a), 1), but now we don't have that problem!!

However, to me it still seems that the neighborhood in the unit circle [1,e^(i*2pi*c)) will be a not closed map whose preimage will be [0,c) U {1} which is $

**$ open either....**

__NOT__Okay so I made the word NOT all fancy because I realized as I was writing this that it was not open because i'm now including the singleton {1}, but i'm going to post this anyway so ya'll can look at my thoughts and give me some feedback as to if my thinking is correct or what not :D