Discuss the Common Core Standards Initiative and the relationship to topology:

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Discuss basis in linear algebra. Discuss a basis for a topology

Which of the following are not part of the definition of a metric?

a) d(x,x)=0

b) d(x,y)=d(y,x)

c) there exists x,y so that d(x,y)=1

d) d(x,z) = d(x,y) + d(y,z)

e) more than one answer is not a part of the definition of a metric clicker test 1

Discuss a variety of metrics including the Chebeychev chess metric, a post office metric, and a game theory metric. Continue Euclidean, taxicab and square metrics and look at balls in them, as well as an open ball on the 2-sphere. Look at the definition of open using metric balls and prove that an open interval is open and a closed interval is not open. Look at the definition of continuity in metrics. Prove that the constant mapping is continuous and that the identity mapping is continuous.

Discuss the importance of sets in point-set topology.
Mendelson p. 4 Exercise #2.
A &sube B, B &sube C -> A &sube C.

Prove homework problem #1 related to unions.

Let f:A &rarr B be a function and C &sube A. Prove that C &sube f^(-1)(f(C).
Then give an example to show that equality fails. Finally, what assumption
do we need to make about f - one-to-one, onto, or both - in order to ensure
equality holds? Prove your answer.

If time remains go over the cartesian product and a proof that
Ax(B &cap C) = (AxB) &cap (AxC)

Does the limit of f(x) = sqrt(x) as x->0 exist?

In 1734 Bishop Berkeley wrote the pamphlet:
*The Analyst, or a Discourse Addressed to an Infidel Mathematician, wherein
it is examined whether the object, principles and influences of the modern
analysis are more deduced than religious mysteries and points of faith.*

Pants Konigsberg Bridges

Real analysis and topology.

Samuel Bruce Smith:

One could say that topological spaces are the objects for which continuous functions can be defined.