MA 4953/6953 - Homework















Starred* problems should be completed if you are enrolled in the course at the 6000 level.
I recommend (but do not require) that all students read and attempt these problems. If you make some significant progress, you should turn it in.

Homework 10, due Apr 28 at 11am

Munkres, p177: 1, 3
Munkres, p186: 3, 5, 8
Munkres, p199: 1, 2*

Homework 9, due Apr 16 at 11am

Munkres, p170: 1, 2, 4, 5, 7, 8*, 9

Homework 8, due Apr 2 at 11am

Munkres, p146: 3*
Munkres, p152: 2, 3, 7, 11
Munkres, p158: 4, 8

Homework 7, due Mar 26 at 11am

Munkres, p111: 5, 6, 8, 9
Munkres, p118: 3, 7*
Munkres, p144: 2

Homework 6, due Mar 19 at 11am

Munkres, p100: 6, 8, 10, 11, 13*, 19
Munkres, p111: 3

Homework 5, due Feb 26 Mar 5 at 11am

Munkres p92: 5, 6, 8, 9, 10*
Munkres, p100: 2, 3

Homework 4, due Feb 12 at 11am

Read about the uniform metric on p124. Please note that ω is the cardinality of the natural numbers. Thus, Rω is the Cartesian product of countable many copies of R. You might think of an element as a (Calc 3-style) sequence of real numbers. Then do the following problems:

Munkres p28: 6, 9, 11
Munkres p126: 2, 6

A. For what functions d( x, y ) = f does f such that f( |x - y| ) define a metric? Notice that we showed on hw1 that f( x ) = x2 or x3 does not work.

Homework 3, due Feb 5 at 11am

Munkres p83: 6
Munkres p126: 1, 3, 10, 11*
Munkres p133: 3a

A. Consider the line with two origins. As a set, this is X = R \ {0} ∪ {0+, 0}. If U is an open set in R which does not contain 0, then U is also open in X. If U is an open set in R which does contain 0, then U \ {0} ∪ {0+} and U \ {0} ∪ {0} are both open sets in X.
Verify briefly that the above really is a topology, then show that X is not metrizable.

Homework 2, due Jan 29 at 11am

Note that a metric subspace is a subset of a metric space taken with the same metric. A metric subspace is itself a metric space. This may be helpful for problems A-D.

Munkres p83: 3, 8

A. Give an example of a continuous bijection between metric spaces which is not a homeomorphism.

B. Find a metric space M and a point x such that M is homeomorphic to M \ x.

C. Prove that the reals R and R \ {0} (both under the Euclidean metric) are not homeomorphic.
Hint: The latter has sets which are both open and closed. Why does this cause trouble?

D. Prove that if f : MN is a homeomorphism of metric space, then for any x in M we have f : M \ {x} → N \ {f(x)} to be a homeomorphism of the given metric subspaces.

Homework 1, due Jan 22 at 11am

Read Sections 1-7. Convinced yourself that you know the answers to Problem 1.2 from the textbook (you need not turn this in). Then answer the following problems, to be turned in:

Munkres p20: 1, 2, 3
Munkres p51: 5*

A. Which of the following determine a metric on the reals?
     i) d( x, y ) = |x - y|2.     ii) d( x, y ) = |x2 - y2|.     iii) d( x, y ) = |x - y|3.

B. Prove that for a function f from a metric space (M1, d1) to a metric space (M2, d2), TFAE:
     i)   f is continuous.
     ii)  If A is open in M2, then the preimage of A is open in M1.
     iii)  If A is closed in M2, then the preimage of A is closed in M1.
     iv)  If an is a sequence in M1 that converges to pM1, then the sequence f( an ) converges to f( p ).

C. Show that if f, g are continuous functions from metric space M to the reals (with the Euclidean metric), then every linear combination of f and g is also continuous.