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Kevin_H
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The book I am using for my Introduction to Topology course is Principles of Topology by Fred H. Croom.
Problem: Prove that if ##X=X_1\times X_2## is a product space, then the first coordinate projection is a quotient map.
What I understand:
To show ##\pi_1:X\rightarrow X_1## is a quotient map, I need to show that (1) ##\pi_1## is surjective, (2) ##\pi_1## is continuous, and (3) for some ##U_1\subset X_1, \pi_1^{-1}(U_1)## open in ##X \Rightarrow U_1## is open in ##X_1##.
##\pi_1## is clearly onto since ##\forall x\in X, \exists x_1\in X_1: \pi_1(x)=x_1.## Let ##O_1## be open subset of ##X_1##. Then ##\pi_1^{-1}(O_1)=\{x=(x_1,x_2)\in X:x_1\in O_1\}=O_1\times X_2##. Thus ##\pi_1^{-1}(O_1)## is open in ##X##. Thus ##\pi_1## is continuous. The third part is what I am not sure of. Since ##\pi_1## is surjective, we have ##\pi_1(\pi_1^{-1}(V_1))=V_1, \forall V_1\subset X_1##. Assume now that ##W=\pi_1^{-1}(O_1)## is an open subset of ##X##. My question is, how would I conclude that ##\pi_1(W)=O_1## is indeed open. Can I just assume or is there something I am missing? I'm sure I am missing something. Out of curiosity, could I generalize this and say if ##X=\prod_1^nX_i## if a finite product space, then the first coordinate projection is a quotient map? How about the ##i##th coordinate projection? I want so say yes. I can easily prove that ##\pi_i:X\rightarrow X_i## is continuous using the same method I did for continuity above. Anywho, that's probably a different story for a different time. Sorry for the long read. If I need to clarify on anything I presented, let me know. I sincerely thank you for taking the time to read this post. I greatly appreciate any assistance you may provide.
Problem: Prove that if ##X=X_1\times X_2## is a product space, then the first coordinate projection is a quotient map.
What I understand:
- Let ##X## be a finite product space and ## x=(x_1,x_2,...,x_n)## a point in ##X##. The function ##\pi_i:X\rightarrow X_i## defined by ##\pi_i(x)=x_i## is called the projection map on the ##i##th coordinate.
- Let ##X## and ##Y## be two topological spaces and ##f:X→Y## a surjective map. The map ##f## is said to be a quotient map provided a subset ##U## of ##Y## is open in ##Y## if and only if ##f^{−1}(U)## is open in ##X##. Equivalently, subset ##U## of ##Y## closed in ##Y## if and only if ##f^{−1}(U)## closed in ##X##.
To show ##\pi_1:X\rightarrow X_1## is a quotient map, I need to show that (1) ##\pi_1## is surjective, (2) ##\pi_1## is continuous, and (3) for some ##U_1\subset X_1, \pi_1^{-1}(U_1)## open in ##X \Rightarrow U_1## is open in ##X_1##.
##\pi_1## is clearly onto since ##\forall x\in X, \exists x_1\in X_1: \pi_1(x)=x_1.## Let ##O_1## be open subset of ##X_1##. Then ##\pi_1^{-1}(O_1)=\{x=(x_1,x_2)\in X:x_1\in O_1\}=O_1\times X_2##. Thus ##\pi_1^{-1}(O_1)## is open in ##X##. Thus ##\pi_1## is continuous. The third part is what I am not sure of. Since ##\pi_1## is surjective, we have ##\pi_1(\pi_1^{-1}(V_1))=V_1, \forall V_1\subset X_1##. Assume now that ##W=\pi_1^{-1}(O_1)## is an open subset of ##X##. My question is, how would I conclude that ##\pi_1(W)=O_1## is indeed open. Can I just assume or is there something I am missing? I'm sure I am missing something. Out of curiosity, could I generalize this and say if ##X=\prod_1^nX_i## if a finite product space, then the first coordinate projection is a quotient map? How about the ##i##th coordinate projection? I want so say yes. I can easily prove that ##\pi_i:X\rightarrow X_i## is continuous using the same method I did for continuity above. Anywho, that's probably a different story for a different time. Sorry for the long read. If I need to clarify on anything I presented, let me know. I sincerely thank you for taking the time to read this post. I greatly appreciate any assistance you may provide.