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Half Open Interval - Geometry - Exam, Exams of Computational Geometry

This is the Exam of Geometry and its key important points are: Half Open Interval, Half Open Intervals, Product Topology, Countable and Separable, Completely Regular, Neighborhood, Arbitrary Product, Regular Space, Proper Map, Topological Manifold

Typology: Exams

2012/2013

Uploaded on 02/14/2013

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GEOMETRY/TOPOLOGY QUALIFYING EXAM
August 2008
INSTRUCTIONS:
You must work on all problems below.
Use a separate sheet of paper for each problem and write only on
one side of the paper.
Write your name on the top right corner of each page.
Problem 1. Let R`be the real line Rwith the lower limit topology, gen-
erated by half-open intervals [x,y ) (also called the Sorgenfrey line). Recall
that a space is called Lindel¨of if every open cover has a countable subcover.
a. Prove that R`is first countable, Lindel¨of, and separable, but not second
countable.
b. Let R2
`be the plane R2with the product topology R`×R`. Show that
R2
`is first countable and separable, but not Lindel¨of.
Problem 2. In this problem all spaces are assumed to be T1.
a. Fix a subbasis Sof X. Prove that Xis completely regular (T31
2) if and
only if for each point xXand neighborhood VSwith xV,
there exists a map f:X[0,1] such that f(x) = 0 and f(y) = 1 for
yXV.
b. Show that the arbitrary product of completely regular spaces is com-
pletely regular. (Hint: Use a.)
c. Show that any subspace of a completely regular space is completely reg-
ular.
Problem 3. Recall that g:XYis called a proper map if g1(C) is
compact whenever CYis compact. Show that if a map f:XYis
closed and f1(y) is compact for all yY, then fis proper.
Problem 4. Atopological manifold Mis a Hausdorff and second countable
space which is locally homeomorphic to an open subset of an Euclidean
space. Show that a topological manifold is metrizable. Is Mparacompact?
Is Mnormal?
Problem 5. Given uRnand cR, define
S:= {(x,y)Rn×Rm| hx,ui2=||y||2+c},
where hx,uiis the inner product of xand u, and ||y|| is the norm of y.
For which constants cis Sa smooth submanifold of Rn×Rm? Prove your
assertion.
Problem 6. Let Xand Ybe smooth manifolds.
a. Show that T(x,y)(X×Y) = TxX×TyY.
b. Define the tangent bundle T(X) of X. Describe how T(X) is given the
structure of a smooth manifold.
August 2008 Geometry/Topology Qualifying Exam Page 1 of 2
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GEOMETRY/TOPOLOGY QUALIFYING EXAM

August 2008 INSTRUCTIONS:

  • You must work on all problems below.
  • Use a separate sheet of paper for each problem and write only on one side of the paper.
  • Write your name on the top right corner of each page.

Problem 1. Let Rbe the real line R with the lower limit topology, gen- erated by half-open intervals [x, y) (also called the Sorgenfrey line). Recall that a space is called Lindel¨of if every open cover has a countable subcover. a. Prove that R is first countable, Lindel¨of, and separable, but not second countable. b. Let R^2 be the plane R^2 with the product topology R × R. Show that R^2 is first countable and separable, but not Lindel¨of. Problem 2. In this problem all spaces are assumed to be T 1. a. Fix a subbasis S of X. Prove that X is completely regular (T 3 1 2

) if and only if for each point x ∈ X and neighborhood V ∈ S with x ∈ V , there exists a map f : X → [0, 1] such that f (x) = 0 and f (y) = 1 for y ∈ X − V. b. Show that the arbitrary product of completely regular spaces is com- pletely regular. (Hint: Use a.) c. Show that any subspace of a completely regular space is completely reg- ular. Problem 3. Recall that g : X → Y is called a proper map if g−^1 (C) is compact whenever C ⊂ Y is compact. Show that if a map f : X → Y is closed and f −^1 (y) is compact for all y ∈ Y , then f is proper. Problem 4. A topological manifold M is a Hausdorff and second countable space which is locally homeomorphic to an open subset of an Euclidean space. Show that a topological manifold is metrizable. Is M paracompact? Is M normal? Problem 5. Given u ∈ Rn^ and c ∈ R, define S := {(x, y) ∈ Rn^ × Rm^ | 〈x, u〉^2 = ||y||^2 + c}, where 〈x, u〉 is the inner product of x and u, and ||y|| is the norm of y. For which constants c is S a smooth submanifold of Rn^ × Rm? Prove your assertion. Problem 6. Let X and Y be smooth manifolds. a. Show that T(x,y)(X × Y ) = TxX × TyY. b. Define the tangent bundle T (X) of X. Describe how T (X) is given the structure of a smooth manifold.

August 2008 Geometry/Topology Qualifying Exam Page 1 of 2

Problem 7. Classify all surfaces with both Gauss curvature and mean curvature equal to zero. (Provide a proof.)

Problem 8. Consider the surface of revolution obtained by rotating the curve t 7 → (0, cos t, sin t) , 0 < t < π/ 2 about the line y = 2. Identify the image of the Gauss map. Problem 9. Let S be a compact surface in R^3. a. Show that S has a point of positive Gauss curvature. b. Prove or give a counterexample: S cannot be a minimal surface.

Problem 10. True or false (answer must be justified): A differential 1- form ω defined on M := R^2 (1, 0) such that dω = 0 and for which there does not exist a smooth function f : M → R with df = ω cannot be extended smoothly to define a differential form on R^2.

August 2008 Geometry/Topology Qualifying Exam Page 2 of 2