Unformatted text preview: x ∈ X and every closed set A ⊂ X with x / ∈ A , there is a function f j ∈ F with f j ( x ) > 0 and f j ( a ) = 0 for all a ∈ A In the next two problems you will need to know that the M¨obius band M is the quotient space obtained from [0 , 1] × [1 , 1] by identifying (0 ,t ) with (1 ,t ). Let q : [0 , 1] × [1 , 1] → M be the quotient map. 4. (a) Deﬁne what it means to say that the subspace A is a deformation retract of the space X . (b) Notice that the image q ([0 , 1] × 0) is a circle C 1 in M . Show C 1 a deformation retract of M . 5. (a) Deﬁne what it means to say that the subspace A is a retract of the space X . (b) Notice that the image q ([0 , 1] ×{ 1 ,1 } ) is also a circle C 2 in M . Is C 2 a retract of M ? Prove your answer is correct....
View
Full Document
 Spring '11
 NA
 Logic, Topology, Continuous function, Topological space, real valued functions, Tube Lemma

Click to edit the document details