MUNKRES TOPOLOGY HOMEWORK
The Hausdorff distance on subsets of a metric space Problem For further advice on writing your homework and project paper , see: Well-Ordering Chapter 2 Section Covering Spaces Section There will be some emphasis on material covered since the first exam.
Our primary goal this semester will be to get through the first four chapters of Munkres’ book. Review of the Basics Chapter 9 Section Injections, surjections, and bijections. We will also study many examples, and see some applications. Open balls and open sets in metric spaces. More about the quotient topology:
We believe, however, that most people will get the maximum benefit from the homework if they try hard to do all the munkrez themselves before consulting others. Also, in Theorem Future topics are tentative and will be adjusted as necessary.
Continuous functions between metric spaces given hmework the epsilon-delta definition. Operations on topological spaces: Furthermore, if f was a continuous map between spaces, then bar f becomes continuous with respect to the quotient topology on.
Unions of subsets which are each connected in homeowrk subspace topology and which have non-empty intersection remain connected. Textbook Topology2ed, by James R.
Munkres (2000) Topology with Solutions
A subset of R equipped with the subspace topology is connected if and only if homewogk is an interval. Finite Sets Section 7: No late exams will be accepted.
Here is the exam. Compact Subspaces of the Real Line Section This means you tpology try to use complete sentences, insert explanations, and err on the side of writing out “for all” and “there exist”, etc. The exam will have a total of six problems, of which students will be required to complete any four.
The closed interval [0,1] is compact. A function between metric spaces is continuous if and only if the preimage of every open set is open. For more details, see the DSP web site here ; in particular contact information is here. The main result we covered is discussed in Munkres 2.
In order to accomodate exceptional situations such as serious illness, your topoligy homework score will be dropped at the end of the semester. An introduction to compactness. Completed proof that products of compact spaces are compact. Hutchings’ Introduction to mathematical arguments including a review of logic and common types of proofs. Sequences and convergent sequences in a metric space. Indexed families of sets and operations on them: Submit final draft to Instructor and Viktor.
The grading will be based on the homework and the take-home examinations. The relationship of connectedness with the notion of topologu separation. Basis for a Topology Section This may involve collaboration with other students.
The Order Topology Section Late homeworks will not be accepted. The Subspace Topology Section Closed Sets and Limit Points Section Homework 4 is due Monday, September No matter how well you think you understand the material presented in class, you won’t really learn it until you do the problems.
Homework 6 is due Wednesday, October 7.