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# Topology I -- Fall 2013 -- final exam

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8 problems, 10 points each

- Show enough work to justify your answers. Refer to theorems whenever necessary.
- One page per problem.

- Consider these subsets of the plane: Q W E R T Y U I O P [ ] \. Think of them as if they are made of wires so that there is no width in these lines. Classify them "up to homeomorphism". (Just the answer)
- (a) State the definition of path-connectedness. (b) Use part (a) to prove that a convex subset of ${\bf R}^N$ is path-connected.
- A topological space is called "Hausdorff" if we can separate any two points from each other by means of disjoint open sets: for any $x,y \in X, x \neq y$, there are open sets $U, V$ such that $x \in U, y \in V$ and $U \cap V = \emptyset$. Are the spaces below Hausdorff?
- anti-discrete space,
- discrete space,
- the real line ${\bf R}$ equipped with the "ray topology": $\{(p, \infty ) \colon p \in {\bf R} \}$.

- For the simplicial complex given below (a) provide two different orientations and (b) compute the boundary operator for each:
- $N=\{A,B,C,D,AB,BC,CD,DA,BD\}.$

- (a) State the $\epsilon-\delta$ definition of continuity of a function at a point. (b) Use part (a) to prove that $f(x)=x^2$ is continuous at $x=0$.
- (a) State the definition of a chain map and provide the main property of chain maps; (b) Use part (a) to prove that chain maps take cycles to cycles.
- (a) Find a simplicial (self-)map a realization of which is a non-hollow triangle being rotated clockwise. (b) Find the chain maps of this simplicial map.
- Two chain complexes and the chain maps of a simplicial map are given below. Find the homology maps.
- $C_0(K)=< A >, C_1(K)=0, \partial _1^K=0;$
- $C_0(L)= < A,B >, C_0(L)= < AB >, \partial_1^L=[-1, 1]^T;$
- $f_0(A)=A, f_1=0.$