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Help me finding Quotient topology about Gauss's notation.

bw0young0math

New member
Jun 14, 2013
27
Hello, thanks for your click this page.

Here is my problem.

Let onto function f: (R,U)→Z (U: usual topology.) ,f(x)=[x] (Gauss's notation.)
Find quotient topology on Z of f.



1.I thought that [Since R,0(empty set),(a,b),(-∞, a),(a,∞) are only open set in (R,U), I have to think of finding quotient topology with R,0(empty set),(a,b),(-∞, a),(a,∞) ]

2.If {a_1,...a_n} is finite subset of Z,
then f^-1({a_1,...,a_n})=[a_1,a_1+1)∪...∪[a_n,a_n+1) is not open set in (R,U).
Thus finite subset of Z is not open in Z.



hum..... then how can I find quotient topology on Z about f??

Please help me...:confused:
 

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
Hello, thanks for your click this page.2.If {a_1,...a_n} is finite subset of Z,
then f^-1({a_1,...,a_n})=[a_1,a_1+1)∪...∪[a_n,a_n+1) is not open set in (R,U).
Thus finite subset of Z is not open in Z.
That is right. Now consider $\emptyset \subsetneq U \subsetneq \mathbb{Z}$ and think about the indiscrete topology on $\mathbb{Z}$.