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TopologyPh.D.QualifyingExam Gerard Thompson Mao-PeiTsui Jan20,2007 This examination hasbeencheckedcarefullyfor errors.If you find whatyou believeto be anerror in a question,reportthis to the proctor. If the proctor'sinterpretationstill seemsunsatisfactory to you, you may modify the questionso that in your view it is correctlystated,but not in sucha way that it becomestrivial. If you feel thatthe examinationis on the long sidedo not panic.The gradingwill be adjustedaccordingly. 1 Part One: Do six questions t . True or false: if (X, d) is a metric spacethen d' definedby d(x,y) = ffi, a metric on X. (x,y € )C is also Explain. 2. Let C be an closed subsetof a topological spaceX. Prove that a subsetA c C is relatively closedin C if and only if A is closedin X. 3 . Define compactnessfor a topological space. A collection of subsetsis said to have the flnite intersectionproperty if every finite class of those subsetshas empty intersection. prove that a topological spaceis compact iff every collection of closed subsetsthat has the finite intersectionproperty itself has empty intersection. 4. Lety be a given cover of a topological spaceX. Assumethat for eachmemberA e y,,there is given a continuous map fe : A + Y such that felAoB=fBlAnB for eachpair of membersA and B of y. Then we may deflne a function f : X + y by taking f(x)= fa(x), (if x eAey). Prove that if y rs a finite closedcover of X, then the function / is continuous. 5. Let X = fLpuuXl,be the product of the topological spaces(Xi,)pE1,a dfid with X having the product topology. Prove that the projection ptr : X + X, is an open map from X onto X, for eachp e M. 6. Prove that a path-connectedtopological spaceis connected. TopologyPh.D.QualifyingExam Page2 of 3 January 2007 T.ProvethatiftwoconnectedsetsAandBinaspaceXhaveacofitmonpointp,thenAUBis connected. 8. Prove that every compact set K in a Hausdorff spaceX is closed. 9. A topological spaceis said to be locally compact if every point has a compactneighborhood. Prove that every closedsubspaceof a locally compactspaceis locally compact. 10. LetX be a topological space. LetA c X be connected. Prove that the closuref of A is connected. 11. Prove that if f : X r+ Y is continuousand surjectiveand X is compactand Y Hausdorffthen / is an identificationmap. 12. Prove or disprove: in a compact topological spaceevery infinite set has a limit point. If you cannotanswerthe questionfor a compacttopological spaceanswerit for a metric space. 2 Part TIvo:Do threequestions 1. Let ,Sr= {(*,;9lf +yz -_\ and 5 4= { ( x rx, z , x 3 , x q , x )+1x43 +f i + f i + f i - 1 } . (i) Let A be the antipodalmap on ,SI definedby A(x,/) = (-x,-y). to the identity map on S 1. Show that A is homotopic (ii) Let B be the map on,S4definedby B(x1,xz,x3,x+,xs)= (-xr,-xz,x3,-x4,-xs). Show that B is homotopic to the identity map on ,Sa. 2. Let X and Y be topologicalspaces. (i) Define what it meansfor X and Y to have the samehomotopy type. (ii) A spaceis contractible if it is homotopy equivalentto the one-point space. Prove that X is contractibleif and only if the identity map idx: X r+ Xis homotopicto amap r: X r+ X whose image is a single point. (iii) Supposethat Y c X. Define what it meansfor Y to be a retract of X. (vi) Prove that a retract of a contractiblespaceis contractible. 3. (i) Let,S2be atwo dimensionalsphere.What is zr1(^S2X Pleaseexplainyour answer. (ii) Let X = XrU Xz U & where Xr = {(x,y,e) e R3l*, + 0 - Dz + z2 = ll, X2 -. {(x,y,e)e R3l* + (y + I)' + zz - Ll and X3-{(0,}1 , ) -l 1 S y < 1 } . The graph of X is sketchedbelow. Find q(X). TopologyPh.D.QualifyingExam Page3 of 3 JanuaryZO07 4. Let P be the two-dimensionalreal projective space andTz be the two-dimensionaltorus. (i) What is zr1(P)? Explain your answer. (ii) The spaceP2 can be obtained from the disk D' by identifyitr1 x - -x if llxll = 1. Let p eint(D2;, the interior of D2. Findthe fundamentalgroup of P - ttpl). (iii) Let f : P' t+ Tz be a continuousmap. Show thatf is null homotopic. 5 . It is well-known that the puncturedplane pz - (0,0) has the structureof a topological group with multiplication defined by (", y) . (p, q) = (xp - !e, xe + yp) (induced by multiplication of complexnumbers).Can one definea topologicalgroup structureon pz - {(1,0),(-1,0)}? Explain. 6 . Outline the main points in the constructionof the fundamentalgroup of a topological space.