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# Topology I -- Fall 2013 -- midterm

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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.

- Sketch the realization of the following graph:
- $N=\{A,B,...,H\}$,
- $E=\{a,b,...,l\}$,
- $∂a = A+B,∂b=B+C,∂c=C+D,∂d=D+A,$
- $∂e = B+F,∂f=C+G,∂g=D+H,∂h=A+E,$
- $∂i = E+F,∂j=F+G,∂k=G+H,∂l=H+E.$

- For the graph with nodes and edges given below find the matrix of the boundary operator (short answer):
- $N=\{A,B,C,D\},$
- $E=\{AB,BC,CD,DA,BD\}.$

- For the graph with nodes and edges given below compute the group of boundaries (short answer):
- $N=\{1,2,3,4,5\},$
- $E=\{12,13,14\}.$

- Suppose $a,b$ aren't multiples of each other and $< \cdot >$ stands for span. Describe this quotient algebraically (short answer):
- $< a,b > / < a-b >.$

- Describe algebraically what happens to the group of cycles of a graph when we add an edge, and no new nodes. (a few sentences)
- Show that any set is both open and closed relative to itself. (short proof)
- (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)
- Prove, from the definition, that $\text{Ext}(A)$ does not contain any limit points of $A$. (short proof)