Math 690 Introduction to Algebraic Topology Spring 2004

EXAM 1

7 problems, 10 points each

Instructions:

1. Provide complete proofs (with all the definitions whenever necessary).

2. Write the problems in the given order.

3. Start each problem on a new page.




  1. Prove that for any set MATH $Fr(A)$ is closed, i.e., its complement is open.

  2. Show that any set is both open and closed relative to itself.

  3. Show that if $X$ is a non-empty topological space with the discrete topology, then the only connected sets are the sets of one element.

  4. Prove that $[0,1]$ is connected. (You can use the fact that it is compact.)

  5. Prove that a closed subset of a compact topological space is compact.

  6. Show that the standard Euclidean (disk) basis of $\QTR{bf}{R}^{2}$ is equivalent to the basis of the product topology of MATH

  7. Let MATH (the unit circle) and $(x,y)\sim(-x,-y)$ for all $(x,y)\in X.$ Describe $X/\sim.$

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