725 Assignments (Fall 2010)
- p.38 #19
- p.40 #2.(b)
- p.41 #5
- p.136 #4
- Fourth Ex on p.3 of the Nets and Filters handout.
- Third Ex on p.4 of the Nets and Filters handout.
BONUS Discuss the composition of associations (or derivations): filter
to nets to filters, and net to filter to nets.
- Ex on p.6 of the Nets and Filters handout.
- p.137 #7 and p.138 #8 as modified
- Refer to p.141 #9.
- Give an example to show that Y must be Hausdorff for
the claim to hold.
- Prove the modified claim.
- Prove or construct a counterexample: if the graph is closed, the
function is continuous.
BONUS Determine when a closed graph implies continuity.
- p.142 #19
- p.170 #20
- A topological space is said to be locally Euclidean if
and only if every point x in X has a
neighborhood homeomorphic to an open set in some Euclidean space.
- Check each separation axiom (i = 0,1,2,3,4,5) for a
locally Euclidean space.
- Prove the line with two origins is locally Euclidean.
BONUS Prove the long line is locally Euclidean.
- Express the line with two origins as a quotient space of the disjoint
union of two copies of the Euclidean line. (Be sure to verify that the
topology is correct.)
- By identifying opposite edges of a rectangle (or square), one obtains
three spaces. One is the torus; determine the other two.
- Is a locally Euclidean space first countable? Second countable?
- p.170 #23
- Express the real projective plane as D² attached to S¹.
- p.174 #6
- p.192 #5
- p.193 #17
- p.194 #10
- p.195 #5
- p.196 #7
- Determine whether (a) the line with two origins and (b) Sierpinski space
are path-connected by arguing directly from the definition in each
case. If not path-connected, determine the path-components.
- A compact subset of a Hausdorff space is closed, and a closed
subset of a compact space is compact. Show that neither holds
- p.228 #9
- p.228 #5.(b)
These last three are due by 4:30 pm Wed 15 Dec.
- p.260 #7
- p.261 #4
- p.262 #6
All rewrites of ungraded assignments
and any bonus problems
are due by 3:00 pm Mon 13
Assignments from Fall '08 and prior.
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