Marywood University Department of Mathematics Course Information for Math 425 Topology by Dr. J.

An introduction to point-set topology. Topics include metric spaces, topological spaces, continuous functions, connectedness, compactness, countability and separation axioms, the fundamental group, and the classification of surfaces. If time permits, some attention will also be given to homtopy and isotopy properties of spaces. The textbook is "Topology", 2nd edition, by T. Gamelin and R. Greene published by Dover ISBN 0-486-40680-6. The course grade is determined by two tests, six homework sets, and a final examination. Useful websites are listed at the end of this page. Prerequisite: Math 204 and Math 324. The picture at the right is the Klein bottle. It is the non-orientable surface with Euler characteristic 0. A Klein bottle can be made from a rectangle by identifying the top and bottom edges using the same orientation, but identifying the left and right edges with opposite orientation. The result is a one-sided tube that needs a fourth coordinate to pass through itself. Hence this construction cannot be done in 3-space without introducing a self-intersection but it can be embedded in 4-space. The Klein bottle can also be formed as the connected sum of two real projective planes: K = RP^2 # RP^2

The Klein Bottle cannot be embedded in 3-space.

 Course Outline

Chapter 1 Metric Spaces 1. Open and closed sets 2. Completeness 3. The real line 4. Products of metric spaces 5. Compactness 6. Continuous functions 7. Normed linear spaces Test #1

Chapter 2 Topological Spaces 1. Topological spaces 2. Subspaces 3. Continuous functions 4. Base for a topology 5. Separation axioms 6. Compactness 7. Locally compact spaces 8. Connectedness 9. Path connectedness Test #2

Chapter 3 Homotopy Theory 1. Groups 2. Homotopic paths 3. Fundamental group 4. Induced homomorphisms 5. Covering spaces 7. Homotopic maps

Sample Problems for Test #1

• True/false.
  • The set of rationals is dense in R.
  • The Cantor set is countable.
  • The Cantor set has empty interior.
  • If { U_n } is a sequence of dense open subsets of a complete metric space X, then intersection(U_n ) is also dense in X.
  • Every open cover of a separable space X has a finite subcover.
  • If every sequence in a metric space X has a convergent subsequence, then X is compact.
  • Every uniformly continuous function is also continuous.
  • All subsets of a metric space are either open or closed.
  • A closed set contains all its adherent points.
• Prove that a closed subspace of a complete metric space is complete.
• Let Y be a subset of the metric space X. If x Œ X is adherent to Y, then prove there exists a sequence in Y that converges to x.
• Give an example of:
  • a complete metric space that is not compact.
  • a collection of open sets whose intersection is not open.
  • a collection of closed sets whose union is not closed.
  • a continuous function (with domain and range) that is not uniformly continuous on the domain.
  • a metric space with a nowhere dense subset.
• A family of closed sets { F_a } in a compact metric space X has the property that any finite subcollection of them have nonempty intersection. Prove that intersection(F_a ) is nonempty.
• Define a metric on Rⁿ by d(x, y) = max { | x₁ - y₁| , | x₂ - y₂| , . . . , | x_n - y_n| }. Show that d actually defines a metric on Rⁿ and sketch the unit ball for n = 2.

Sample Problems for Test #2

• True/false
  • If f : X ---> Y is a continuous function and A is open in X, then f(A) must be open in Y.
  • If f : X ---> Y is a continuous function and x is a limit point of the subset A of X, then f(x) is a limit point of f(A).
  • Any open ball in Rⁿ is homeomorphic to Rⁿ.
  • Every metric space is normal.
  • A closed subset of a Hausdorff space is compact.
  • A limit point in any topological space is always unique.
  • Consider Y = [0, 2] as a subspace of R. Then [0, 1) is open in Y but not open in R.
  • A first countable space is always second countable.
  • In a T₁-space, all one-point subsets are closed.
• Define the following terms.
  • Adherent point of a subset S of a space X
  • Separable space
  • Base for a topology
  • Topological property
  • Connected component of a space X
  • Regular space
  • Disconnected space
• Prove that the limit of a convergent sequence is unique in a Hausdorff space.
• State the Urysohn Lemma. If the conclusion of this lemma is true for a space X, then show that X must be normal.
• Prove that a set X with the cofinite topology is compact.
• Prove that an infinite set X with the cofinite topology is not Hausdorff.
• Let X be a set. Is the collection T = { U | X\U is infinite or empty or all of X } a topology for X? Explain.
• A function f : X ---> Y is called closed if f(A) is a closed set whenever A is a closed set in X. If X is compact and Y is Hausdorff, show that a continous function f : X ---> Y is also a closed function.
• Let X = C union D where C and D are non-empty, disjoint, and open. Show that if Y is a connected subset of X, then Y lies entirely in C or Y lies entirely in D.
• Show that the unit circle { (x, y) in R² | x² + y² = 1 } is not homeomorphic to R.
• Prove that connectedness and compactness are topological properties.
• Show that a closed subspace of a normal space is normal.
• Use the notion of a cut point to demonstrate that (0, 1) is not homeomorphic to [0, 1].
• Show that the T₁ property is hereditary.

Useful Websites

http://at.yorku.ca/topology/ (Topology Atlas - Information Related to Topology including preprints, abstracts, who's who, etc. In particular the Education link provides some very good learning resources.)

http://www.math.niu.edu/~rusin/known-math/index/54-XX.html (Introductory Notes to General Topology)

http://www.mapleapps.com/index.asp (The Maple Applications Center. One module here generates finite topologies. Another module explores knot theory.)

http://www-groups.dcs.st-andrews.ac.uk/~history/HistTopics/Topology_in_mathematics.html (A History of Topology)

http://humber.northnet.org/weeks/ (Torus and Klein Bottle Games)

http://www.math.rochester.edu/misc/klein-bottle.html

http://www.geom.umn.edu/zoo/toptype/klein/

Comments to Dr. Craig M. Johnson, Chair, Dept. of Mathematics: johnsonc@ac.marywood.edu

Last update: March 10, 2003

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