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# Topology II -- Spring 2014 -- final exam

Name:_________________________

5 problems, 10 points each

- Justify every step you make with as thorough explanation as possible.
- Unless requested, no decimal representation of the answers is necessary.
- Start every problem at the top of the page.

$\bullet$ **1.**Prove that any convex subset of ${\bf R}^n$ is simply connected.

$\bullet$ **2.** Give the definitions of the chain maps and the homology maps of cell maps. Give an example of two different cell maps with the same homology map.

$\bullet$ **3.** State the Simplicial Approximation Theorem with definitions. Find a simplicial approximation of the rotation of the triangulated circle through $\sqrt{2}\,\pi$.

$\bullet$ **4.** Bolzano-Weierstrass Theorem states that in a compact space, every infinite subset has an accumulation point. Prove that in a compact space, every sequence has a convergent subsequence.

$\bullet$ **5.** Give the definition of homotopy equivalence. Describe and prove the relation between that and homeomorphisms.