There will be some emphasis on material covered since the first exam. Future topics are tentative and will be adjusted as necessary. Homework 7 is due Monday, October In complete generality, compactness and sequential compactness both imply limit point compactness, but compactness and sequential compactness are not equivalent and one doesn’t imply the other. It is recommended that you take at least one other upper level mathematics course before Math i.
Closed sets, and first examples. Chapter 1 Section 1: Unions of subsets which are each connected in the subspace topology and which have non-empty intersection remain connected. Sequences in sets are defined in Munkres p. Homework 9 half weight. Examples of quotient maps of sets coming from partitions , which in turn are often sets of equivalence classes under an equivalence relation.
Homework 8 munjres due Wednesday, October Continuous Functions Section Sequences and convergent sequences in a metric space. Other notions of compactness: Infinite Sets and the Axiom of Choice Section Clark Fractals and Self Similarity by J.
Math 440: Topology, Fall 2017
The definition of the fundamental group. More about continuity being equivalent to sequential continuity. An introduction to compactness. Every function from a discrete metric space is continuous.
A more detailed lecture plan updated on an ongoing basis, after each lecture will be posted below. Students are not allowed to work together on these.
Retractions and Fixed Points Section This means you should try to use complete sentences, insert explanations, and err on the side of writing out “for all” and “there exist”, etc. The interior of a set. Homework Homework 1 is due Monday, August Normal Spaces Section Examples of metric spaces: In complete generality, compactness and sequential compactness both imply limit point compactness, but compactness and sequential compactness are not equivalent and one doesn’t imply the other.
Reading After finishing our discussion of the Arzela-Ascoli-Frechet theorem and the compact-open topology, we will otpology as many of the following topics as the remaining class time allows: Properties of topological spaces: Homework 5 is due Wednesday, September Math will have weekly homework assignments, posted here a week or more before they are due.
Submit first draft to Instructor and Viktor. The instructor strongly adheres to the University policies regarding principles of academic honesty and academic integrity violations, and will strictly enforce these rules. Homework 2 is due Wednesday, September 9.
Math Introduction to Topology I
You may work with others and consult references including the course textbookbut the homework you turn in must be written himework you independently, in your own language, and you must cite your sources and collaborators.
While the material for the final exam will cover the entire course, there will be somewhat of an emphasis on the material covered after the prelim.
Closed Sets and Limit Points Section Concretely, with a small group of students, you will be asked to write a short expository article around pages, typedand give an in-class 20 min presentation. More about abstract topological spaces.
Below are links to answers and solutions for exercises in the Munkres Topology, Second Edition. Munkres Topology with Solutions. A review of one way to construct functions out of bar X. Textbook Topology2ed, by James R.
Time permitting, we may include other topics, such as the fundamental munkrse of a topological space.