Question

Let X : = Π_α∈IX_α be a product space (with the product topology), π_α : X → X_α be the projection map for each α∈I, and {x_n}$_{}^{}$$_{n=1}^{\infty }$ be a sequence in X. Prove that the sequence {x_n} converges to a point x∈X if and only if {π_α(x_n)} converges to π_α(x) for every α∈I.

Answer 1

Ans Given x := IT Xx is a product space LEI To: X → X is the projection map for each LEI. Now {xnn is دن a sequence in X. Let the sequence {xn] be convergess to the pt x EX let LEI. ana U Ç Xa be on open nba of Ita (2). let Vis IT U; & IT Xi it I it I where x = u and U; = x; for iza Then vis an open nba of 2 Hence almost all xn E V > almost all Ita (xn) EU. Ita (xn) converges to Ta (2) This true for all LEI. {ita (Xn)} converges to M(x) for every a EI {ta (xn)} converges to M(x) for conversely let every LEI. Let be any open nba of x in X. Zhen a basic open set v such that x Ev su lchere V=IT V 2 Y = xx sor at 2,62,...an

Alow ki open nba for each Xi (@i 21,2,.. n) is an To(x) in Xa; by hypothesis { a (xn} converges to Mag(x) i=1,2,...,n ans in Xxi U 7 Ki EN s.t. T. (Rn) € Vai * na, ki det ka man دم K, ,kg,.. Kn] IT Tai (en) E vai x n >,K., 1= 1,2 r. n. Xn € IT va VCU V n >K. the sequence {un) is {an) is converges to to the ptx en X. the a paint XEX sequence {x} converges to ift { [a (Xn)} converges to IQ(x) for every XEI