clicker test 3
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
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
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.
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).
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.
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.
Thur Sep 8
Go over metric space exercises.
Look at a hyperbolic metric.
Discuss a proof that the square metric is
Tues Sep 6
Which of the following are not part of the definition of a metric?
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.
Fill out index sheets and introductions.
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.
Real analysis and topology.
Samuel Bruce Smith:
One could say that topological spaces are the objects for which continuous
functions can be defined.