# Topology2009jan - x ∈ X and every closed set A ⊂ X with...

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Topology Qualifying Exam January 9, 2009 Do all of the following problems each of which is worth 20 points. 1. Let X be a topological space. (a) Deﬁne what it means to say that X is compact . (b) State the Tube Lemma. (c) Prove the Tube Lemma. 2. Let X and Y be topological spaces and f : X Y be a continuous function. (a) Deﬁne what it means to say that the continuous function f : X Y is closed . (b) Let π 1 : X × Y X be projection on the ﬁrst factor. If Y is compact, show that π 1 is closed. (c) Is the statement in part (b) true if Y is not compact? Show your answer is correct. 3. Let X be a T 1 -space and F = { f j : X R | j J } be a family of continuous real valued functions that separates points and closed sets. Deﬁne F : X R J by F ( x ) = ( f j ( x )) j J . Show that F is an embedding of X into R J . Hint: Recall that a family F = { f j : X R | j J } of continuous real valued functions deﬁned on the space X is said to separate points and closed sets if, for every point
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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....
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## This note was uploaded on 06/19/2011 for the course MATH 661 taught by Professor Na during the Spring '11 term at University of North Carolina School of the Arts.

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