Topology I -- Fall 2013 -- midterm

Name:_________________________

8 problems, 10 points each

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

1. Sketch the realization of the following graph:
   • $N=\{A,B,...,H\}$,
   • $E=\{a,b,...,l\}$,
   • $∂a = A+B,∂b=B+C,∂c=C+D,∂d=D+A,$
   • $∂e = B+F,∂f=C+G,∂g=D+H,∂h=A+E,$
   • $∂i = E+F,∂j=F+G,∂k=G+H,∂l=H+E.$
2. For the graph with nodes and edges given below find the matrix of the boundary operator (short answer):
   • $N=\{A,B,C,D\},$
   • $E=\{AB,BC,CD,DA,BD\}.$
3. For the graph with nodes and edges given below compute the group of boundaries (short answer):
   • $N=\{1,2,3,4,5\},$
   • $E=\{12,13,14\}.$
4. Suppose $a,b$ aren't multiples of each other and $< \cdot >$ stands for span. Describe this quotient algebraically (short answer):
   • $< a,b > / < a-b >.$
5. Describe algebraically what happens to the group of cycles of a graph when we add an edge, and no new nodes. (a few sentences)
6. Show that any set is both open and closed relative to itself. (short proof)
7. (a) Show that the set $\gamma =\{(a,\infty): a\in {\bf R}\}$ of all open right rays is a basis of neighborhoods in ${\bf R}$. (b) Prove that $\gamma$ is not equivalent to the Euclidean basis of ${\bf R}$. (short proof)
8. Prove, from the definition, that $\text{Ext}(A)$ does not contain any limit points of $A$. (short proof)