Download Topology Ph.D. Qualifying Exam Jan 20,2007 Gerard Thompson

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Gerard Thompson
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) =
a metric on X.
(x,y € )C is also
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
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.
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January 2007
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
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 -_\
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
, ) -l 1 S y < 1 } .
The graph of X is sketchedbelow. Find q(X).
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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)}?
6 . Outline the main points in the constructionof the fundamentalgroup of a topological space.