Binghamton University, SUNYMath 461: Topology

Fall 2009


Marco Varisco, [how to email a professor],
Office: LN-2232, Office Hours: MW 3:30–5:00 or by appointment.


MWF 2:20–3:20 in LN-1406 and R 1:15–2:40 in LN-1408.
N.B.: The Thursday meeting is not a discussion session.

You are expected to attend all class meetings. The maximum number of absences permitted to receive credit for this course is 5 (five). Excessive tardiness may count as absence. [University Bulletin]


Math 304: Linear Algebra, and Math 323: Calculus III, and Math 330: Intro to Higher Math; or consent of Department. [University Bulletin]


“Study of topological spaces. Metric spaces, separation properties, connectivity, compactness.” [University Bulletin]


James Munkres, Topology, Second Edition, Prentice Hall, 2000.

Grading & Examinations

Of course, you are expected to obey the Student Academic Honesty Code.


#Due onProblems
1W 9/9 Exercise §2.4.
R 9/10 Retake (but do not hand in) quiz #1.
W 9/16 Retake (but do not hand in) quiz #2.
2R 9/17 In R2 draw the spheres of radius 1 centered at the origin (0,0) with respect to: 1) the Euclidean metric; 2) the taxicab metric; 3) the square metric; and 4) the discrete metric.
3M 9/21 Retake (but do not hand in) quiz #3, and solve the following problems.
A] Let (X,d) be an arbitrary metric space. Prove that for all distinct points x1 and x2 in X there exist positive real numbers ε1 and ε2 such that Bd(x11) and Bd(x22) are disjoint.
B] Find a metric space and two balls in it such that the ball with smaller radius contains and is not equal to the ball with larger radius.
4R 9/24 Let (X,d) be an arbitrary metric space. Given any point a in X and any non-negative real number r, prove that { xX | d(a,x)≤r } is closed.
5F 9/25 Write down explicitly what it means for a function between metric spaces to be discontinuous.
6M 10/5 Retake (but do not hand in) quiz #4, and solve exercise §13.1: Let X be a topological space, and let A be a subset of X. Suppose that for each xA there is an open set U containing x such that UA. Show that A is open in X.
7R 10/8 Prime time!
8M 10/19 Exercises §17.6, §17.7, and §17.8.
R 10/22 Retake (but do not hand in) quiz #6.
9W 11/4 Let (X,d) be a metric space, let A be a subset of X, and let x be a point in X. Show that x is in the closure of A if and only if inf{d(x,a)|aA}=0.
M 11/9 Riddle: consider a set S of disjoint figure-eight curves in the plane; can S be uncountable?
10M 11/16 A] Show that the lower limit topology on R is Hausdorff and first-countable, but not metrizable. (Hint: in order to prove that it is not metrizable, show that it is … but not …, and apply a result discussed in class.)
B] Show that any set of pairwise disjoint open subsets of a separable topological space is countable.
C] Show that any discrete subspace of a second-countable topological space is countable.
R 11/12 Retake (but do not hand in) quiz #7.
11M 11/23 In parts B] and C] below, the symbol R denotes (as usual) the real numbers with the standard topology.
A] Suppose that X and Y are topological spaces and that fX → Y is a continuous function. Show that if X is compact then f(X) is compact.
B] Suppose that C is a non-empty compact subspace of R. Show that C has a least element, i.e., that there exists cC such that for all xC we have cx. (Hint: argue by contradiction, and observe that { (x,∞) | xC } is an open cover of C if and only if C does not have a least element.) Is c unique?
C] Suppose that X is a non-empty compact topological space and that fX → R is a continuous function. Use parts A] and B] to show that there exists mX such that for all xX we have f(m)f(x). Is m unique?
12M 11/30 Study section §23 up to and including theorem 23.5 (pages 148–150), and solve exercises §23.1, §23.2, §23.5, and §23.7 (recall that the symbol <em>R<sub>l</sub></em> in §23.7 denotes the real numbers with the lower limit topology).
13M 12/7 Exercises §24.3, §24.8(c), and §24.8(d).
M 12/7 Retake (but do not hand in) quiz #9.

This syllabus is subject to change. All official announcements and assignments are given in class, and this web page may not be up to date.
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