MATH418 - Topology

After successfully completing this course, you will be able to

CO-1 Determine whether a collection of subsets of a set is a topology and determine a basis for a topology.
CO-2 Use homeomorphisms and show two spaces are topologically equivalent.
CO-3 Explain and proof the basic properties of compactness and of connectedness.
CO-4 Determine if a topological space is a metric space and generate a topology from a metric.
CO-5 Use the subspace topology, the product topology, and the quotient topology.
CO-6 Explain and proof basic separation axioms and properties of Hausdorff spaces.

Reading Assignments: Reading assignments are provided each week. These assignments flow into the Forum discussions and homework problems. Reading assignments are not graded directly; however, required homework problems must be submitted via the Messages by Sunday at midnight. Optional homework problems are for your benefit and are not required for submission. Your conceptual understanding, ability to solve problems, and ability to synthesize material will be evaluated using quizzes, projects, and a final exam.

Forum Assignments: Mathematics is not a spectator sport. In order to learn the language of Mathematics, you must be engaged with the material. It is critical that you spend time thinking, considering examples, working problems, and discussing ideas with others. The Forums are evaluated in three areas: quantity of posts, quality of posts, and value of interactions.

Quantity – The initial post for each Forum includes at least 250 words, and a minimum of two interaction posts are required per Forum using at least 100 words each.
Quality – High quality posts are critical to the development of everyone in the course. The overall quality of your posts is evaluated.
Value – Banal posts such as “Good work” and “Nice conclusion” provide no value to the Forum conversations. The key to the Forums is quality interaction. Superior posts promote a valuable conversation and meaningful interaction.

Evaluation criteria	Descriptive adjectives	Scoring
Quantity	First post (>250 words) Two interaction posts (>100 words each)	33%
Quality	Accuracy, logical presentation, organization, clarity, completeness, proper terminology	34%
Value	Contributing to the conversation, useful to your colleagues, valuable feedback	33%

Forum Note: you cannot score points for the quality and value of a post if you fail to meet the minimum quantity.

Writing Assignments: Written communication is a key piece of modern mathematics. The Forums as well as many of the homework problems ask you to develop an argument and write it clearly. In addition to the Forums and homework, you are required to write two formal proofs as writing assignments. These assignments are evaluated according to their validity, readability, and fluency. The definitions for those concepts are given here.

Validity – Validity corresponds to the validity of your arguments. It addresses the extent to which your method is appropriate, your calculations are correct, and your deductions follow the rules of logic.
Readability – If your written work is not readable it cannot be assessed. Since the ability to communicate Mathematics is a focal point for this class, special attention will be paid to the readability of your work.
Fluency – Mathematics is a concise and precise language, and I wish to enhance your fluency. Therefore, part of every assessment will focus on your ability to incorporate correct, established notation and terminology into your written work.

Evaluation criteria	Descriptive adjectives	Scoring
Validity	logical arguments, deductive reasoning, proper conclusions, complete reasoning	43%
Readability	organization, presentation, format, clarity, effectiveness	35%
Fluency	proper notation, proper terminology, appropriate definitions, conciseness	22%

Quizzes: These are the core assessment tools for the assigned readings and homework. Your work will be graded for correctness, completeness, and clarity.

Multimedia Presentation: As with the writing assignments, the multimedia presentation provides you with an opportunity to improve your Mathematical communication. For the presentation you will need to effectively communicate your ideas in a slideshow presentation that includes an audio track. Using audio files you will explain your work with clarity and precision. This assignment will be evaluated in a manner similar to the writing assignments. However, an additional criteria will be evaluated.

Multimedia – A completely integrated presentation incorporating written work with emphases and accompanying audio. The total effect of the presentation is evaluated.

Evaluation criteria	Descriptive adjectives	Scoring
Validity	logical arguments, deductive reasoning, proper conclusions, complete reasoning	34%
Readability	organization, presentation, format, clarity, effectiveness	27%
Multimedia	incorporated audio, highlights, emphases, audience engagement, integration	22%
Fluency	proper notation, proper terminology, appropriate definitions, conciseness	17%

Final Exam: As with the quizzes, the final exam is a core assessment tools for the assigned readings and homework. The final exam is comprehensive, and your work will be graded for correctness, completeness, and clarity.

Please see the Student Handbook to reference the University’s grading scale.

Name	Grade %
Forums	23.00 %
APUS Honor Code	0.96 %
Introduction Forum	0.96 %
Week 1 Forum & Homework	2.88 %
Week 2 Forum & Homework	2.88 %
Week 3 Forum & Homework	2.88 %
Week 4 Forum & Homework	2.88 %
Week 5 Forum & Homework	2.88 %
Week 6 Forum & Homework	2.88 %
Week 7 Forum & Homework	2.88 %
Week 8 Feedback	0.96 %
Writing	26.00 %
Writing 1	8.00 %
Writing 2	8.00 %
Presentation	10.00 %
Quizzes	30.00 %
Quiz 1	10.00 %
Quiz 2	10.00 %
Quiz 3	10.00 %
Final Exam	21.00 %
Final Exam	21.00 %

Additional Resource
Topology Without Tears by Sidney A. Morris
Version of 3 March 2013
Online textbook: http://www.topologywithouttears.net/topbook.pdf

Web Sites
In addition to the required course texts, many public domain web sites are useful. Here are a few sites to consider if you want to view topology and modern mathematics from a different perspective. Please abide by the university’s academic honesty policy when using Internet sources. Note web site addresses are subject to change.

Site Name … youtube.com	Web Site URL/Address
Intro to Topology by Franzosa	http://www.youtube.com/watch?v=zsN_guq Ac
Topology basics	http://www.youtube.com/watch?v=tWIoUZNYj1g
Topology basics	http://www.youtube.com/watch?v=cBMtwB8xoTE
Intro to Topology	http://www.youtube.com/watch?v=Z-4Uo1pv0mU
Neighborhoods	http://www.youtube.com/watch?v=6F3iJ1utuG0
Int, Cl, Bdry part 1	http://www.youtube.com/watch?v=4N-DS2xxZAY
Int, Cl, Bdry part 2	http://www.youtube.com/watch?v=7e4zC-PLZxI
Higher Math – Infinity	http://www.youtube.com/watch?v=RZL5bKk9IHs
Higher Math – Real numbers	http://www.youtube.com/watch?v=YW3EDsmj9bo
Higher Math – Topology	http://www.youtube.com/watch?v=FHL4udeLf9Q
Higher Math – Sequences	http://www.youtube.com/watch?v=rMe8YaoyDBA
Metrics	http://www.youtube.com/watch?v=QDz_D2BPVG0

Book Title:	Introduction to Topology: Pure and Applied - the VitalSource e-book is provided inside the classroom
ISBN:	9780131848696
Publication Info:	VS-Pearson
Author:	Adams, Colin
Unit Cost:	$181.57
Electronic ISBN:	9781269653152
Electronic Unit Cost:	$37.80