Question

Let S 1 be the unit circle with the usual topology, S 1×S 1 be the product space, and define the torus T := [0, 1] × [0, 1]/ ∼ as a quotient space, where ∼ is the equivalence relation that (1, y) ∼ (0, y) for all y ∈ [0, 1] and (x, 0) ∼ (x, 1) for all x ∈ [0, 1]. Prove that S 1 × S 1 and T are homeomorphic.

(20 points) Let Sl be the unit circle with the usual topology, Sł x Sl be the product space, and define the torus T [0, 1] x [0, 1]/ ~ as a quotient space, where ~ is the equivalence relation that (1, y) ~ (0,y) for all y € [0, 1] and (x,0) ~ (x,1) for all x € [0, 1]. Prove that S1 x S1 and T are homeomorphic.

Answer 1

SOU Given that Let s' be the unit circe with the usual topology e s'xs' be the procłuct space, and define the torus 7: =[001]*[0,1]/m as a quobrent space, where an is the cruivalent relation that (2.7) ~ Cory for all y E[0,1] and [2.0) (9.1) for QUI TE (0,2) Take x = IXI where Ic[0, 1] and par bhon xoto 1) The set (10,0),(0,1), Clio),(1,1)} (four corner points) 9) set converting o pay of points (2.0), (, 1) JE 1 3) set converting of pair of points cory), (2,4), YE 2 4) set converting of a single point (x, y) where 그 xiy Ecoil) Then the stesuiting quotent (Ideal flaBon space) in the Torus If we take s'cc, and detine fisxs -> TXI 22TTIM Qenly) (rigs (e2min Now loot at f'(2) for ZE S xs Trey Parb?Bon IX I and they are the sett mentioned above f is identifcation map If filmy is a poto continous function, where a is compact and an identlb'cabon map] The two s*x s? and Torus are hemeomorphis Y ?s trass dortt, then fis