Tues Nov 22 Continue compactness. Discuss discrete topology. Show that [0,1] is compact. Proofs related to compactness.

Thur Nov 17 Test 2

Tues Nov 15 Discuss path connected. Prove that path connected spaces are connected but that the reverse is not true [flea and comb space]. Begin compactness. Examine R, (0,1), R_l, X with finitely many opens.

Thur Nov 13 Go over the exercise set. Prove that the product of connected spaces is connected. review clicker questions.

Tues Nov 11 Continue connectedness. Prove that a continuous surjection preserves connectedness. Prove that the union of connected spaces with nonempty intersection is connected. Connectedness by points. Discuss the product of connected spaces, and cubes, cylinders, etc...

Thur Nov 3 Finish Hausdorff and begin connectedness.

Tues Nov 1 Prove that X is metrizable then X is Hausdorff. Prove that Xcf (X inf) is not Hausdorff and conclude that Xcf (X inf) is not metrizable. Prove that continuity iff closed statement. Composition of continuous functions. Piecewise continuous functions and product of continuous functions. Review bijection. Definition of homeomorphism. Look at f(x) = x/(1-x^2). Compare Rl and R and prove that homeom preserves Hausdorffness.

Thur Oct 27 Go over the exercise set. Discuss Rl as a space that is Hausdorff but not metrizable. Revisit f(x)=x^2. Look at the image of (-1,1), the inverse image of [0,1) and (-.5,1) as well as a pictorial proof that the inverse image of basic opens in R go back to opens in R. Prove that continuity in real analysis implies continuity in topology

Tues Oct 25 Look at the first 3 Clicker questions on continuity Look at g(x)=x from Rl-->R in terms of continuity. Begin Hausdorff.

Thur Oct 20 Set theory fill in the blank proofs. Continue continuity, including maps from the discrete topology or to the indiscrete topology. Look at f(x)=x from R-->Rl and g in reverse.

Tues Oct 18 Review and finish boundary points. Begin continuity. Review real-analysis definition and the set theory definition of inverse image and direct image. Topological definition. Rewrite the standard definition in terms of preimage: (x0-delta,d0+delta) subset inverseimage (f(x0)-epsilon, f(x0)+epsilon). Then

Thur Oct 13 Interior, Closure and Limit Points. Boundary.

Tues Sep 19 Review the discrete topology, standard topology and cofinite topology on the reals, and compare with the indiscrete topology on R. Clicker questions Review generating topologies: finite intersections and arbitrary unions. Revisit open in a topology. Define closed (complement is open). Take questions on LaTeX. Hand out sheet with a proof that the topology on R² induced by the Euclidean metric is the same as the topology induced by the square metric. Show that R_k is not comparable to R_l. Look at generating topologies using lower and upper limit basis elements. Look at generating a topology with the complement of one point. Discuss the Digital topology and the phenotype topology.
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

Thur Sep 15 Discuss the definition of a topology and why a metric space satisfies the definition. Examples and non-examples of topologies, including the Serpinski topology, the cofinite topology, as well as the indiscrete topology, the discrete topology and the standard topology on the reals. Comparing topologies. Compare the discrete topology to the standard topology and the cofinite topology on the reals.

Tues Sept 13 Prove that the intersection of opens is not necessarily open. Prove that open in R equals union of open intervals. Clicker questions.

Thur Sep 8 Go over metric space exercises. Look at a hyperbolic metric. Discuss a proof that the square metric is a metric.

Tues Sep 6
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.

Day 4: Thur Sep 1: Try to prove that f(x)=|x| is continuous at x_0 given that ||x|-x0|| leq |x-x0| and also discuss a proof by cases. Highlight the importance of the triangle inequality in the proof of the lemma. Discuss metric spaces in both books. Look at the Euclidean metric, the discrete metric and the taxicab metric and run through the metric axioms.

Day 3: Tues Aug 30 Go over Set Theory Exercises 2. Discuss the material from 2110 and 2510. Discuss both books presentation of set theory. Finish proving homework problem #1 ???

Day 2: Thur Aug 25 Register the i-clickers. Then sit in chairs facing the other way. Collect hw. Go over 1st 2 problems via clickers and then call on students to present. Discuss what elements when into the Intermediate Value Theorem. Discuss a commutative diagram representing IVT that will lead to topological concepts including connectedness.

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)

Day 1: LaTeX slides Fill out index sheets and introductions.
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.

I have no controversy about your conclusions, but only about your logic and method. How do you demonstrate? What objects are you conversant with, and whether you conceive them clearly? What principles you proceed upon; how sound they may be; and how you apply them?

Pants

Leonhard Euler:

[upon losing the use of his right eye] Now I will have less distraction. Quoted in H Eves In Mathematical Circles (Boston 1969).

If a nonnegative quantity was so small that it is smaller than any given one, then it certainly could not be anything but zero. To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be. These supposed mysteries have rendered the calculus of the infinitely small quite suspect to many people. Those doubts that remain we shall thoroughly remove in the following pages, where we shall explain this calculus.

Konigsberg Bridges

Real analysis and topology.

Samuel Bruce Smith:

The axiomatic rigor that makes topology a model and solid foundation for other fields is precisely the characteristic that makes it a difficult fit for the undergraduate curriculum.... One of the great advantages of topology is the almost visual elegance of its formalism.

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