- #1
PsychonautQQ
- 784
- 10
Hey PF!
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 $NOT$ open either...
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
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 $NOT$ open either...
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