Topology I -- Fall 2013 -- final exam

Name:_________________________

8 problems, 10 points each

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

1. 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)
2. (a) State the definition of path-connectedness. (b) Use part (a) to prove that a convex subset of ${\bf R}^N$ is path-connected.
3. 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?
   1. anti-discrete space,
   2. discrete space,
   3. the real line ${\bf R}$ equipped with the "ray topology": $\{(p, \infty ) \colon p \in {\bf R} \}$.
4. For the simplicial complex given below (a) provide two different orientations and (b) compute the boundary operator for each:
   • $N=\{A,B,C,D,AB,BC,CD,DA,BD\}.$
5. (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 $x=0$.
6. (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.
7. (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.
8. Two chain complexes and the chain maps of a simplicial map are given below. Find the homology maps.
   • $C_0(K)=< A >, C_1(K)=0, \partial _1^K=0;$
   • $C_0(L)= < A,B >, C_0(L)= < AB >, \partial_1^L=[-1, 1]^T;$
   • $f_0(A)=A, f_1=0.$