basic course info - cpb-us-w2.wpmucdn.com€¦ · p goes around each of a, b and c with winding...

basic course info

Class meetings Wed 6:30 – 9:20 in Ball 214

Instructor Prof. Tibor BekeOffice Olney 428-VE-mail Tibor [email protected]

Phone (978) 934-2445Office Hours Wednesday, Thursday 2:00 – 4:00, or by appointment

Textbook Gamelin and Greene: Introduction to Topology2nd edition. Dover Publications, 1999

Prerequisite Real Analysis

Course home page sites.uml.edu/tibor-beke/teaching

Tibor Beke topology: intro

sites.uml.edu/tibor-beke/teaching

what is topology?

The branch of geometry concerned with properties of shapes thatare invariant under continuous deformations.

• What is a shape, exactly?

• Continuous: OK; but what is a continuous deformation?

• Any common sense examples of such properties?


map of London


subway map of London


“ common network topologies ”


the winding number


the winding number does not change under homotopy


theorem!

If two closed paths in the plane have the same winding numberabout the origin, then they are homotopic.

I Visually, sort of, plausible.

I Definitions of path and homotopy : pretty straightforward.

I Precise definition of winding number : not at all easy.

I Proof of the above theorem: fairly difficult.


does the winding number show up in mathematics?

Let α be an oriented, closed path in Cr {0}. Then the complexline integral ∮

α

1

zdz = 2 · wα · πi

where wα is the winding number of α around the origin.


challenge

How can one classify ways of drawing a curve around three points?


Homework Problem 1

Let a, b, c be three distinct points in the plane. Can you drawcurves P and Q in the plane, avoiding the points a, b, c , such that

• P goes around each of a, b and c with winding number +2• Q goes around each of a, b and c with winding number +2• P cannot be continuously deformed into Q (without passingthrough one of the points a, b, c).

Your answer should be ‘yes’ or ‘no’. We do not have themathematical machinery to prove anything at this point, so feelfree to argue intuitively!

Due Wed, Jan 29, in class.


what the cover of your textbook should look like


Homework Problem 2

From your textbook: Chapter 1, Section 1, Exercise 1 on page 7.Parts a), b), c) only.


Start reading your textbook! The above exercise requires noprerequisites other than logical thought.


Tentative Syllabus

I weeks 1-2: metric spaces; axioms, convergence, completeness

I weeks 3-4: metric spaces: open, closed sets; continuous maps

I weeks 5-6: topological spaces; continuity; connectedness

I weeks 7-8: product topology; quotient topology

I weeks 9-10: compactness; Heine-Borel theorem

I weeks 11-12: paths, path lifting, winding number

Final exam On the date and time set by the Registrar. Will coverthe entire semester, and will count for 25% of your grade.

Midterm evaluation One take-home midterm exam, due Wed,March 4. Will account for 20% of your grade.


Homework problems Typically four assigned in each class, duethe following Tuesday. Homework accounts for 45% of your grade.

Email All official communication will be sent to your student emailaddress, Your [email protected].

Class attendance policy You are expected to be present at allclass meetings and take all exams on their assigned dates. If youhave religious or athletic obligations recognized by Universitypolicy, you have to notify me of any class conflicts as soon as youcan, and not less than a week in advance.

Class attendance accounts for 10% of your grade.


what’s out there


three equivalent definitions of continuity

Let f : R→ R be a real-valued function of one real variable.

Definition 1. f is continuous (over its entire domain) if for everyx0 ∈ R and every ε > 0 there exists a δ > 0 such that wheneverx0 − δ < x < x0 + δ then f (x0)− ε < f (x) < f (x0) + ε.




Definition 2. f is continuous (over its entire domain) if for everyconvergent sequence {xi}i∈N one has

limi→∞

f (xi ) = f(

limi→∞

xi).




Definition 3. f is continuous (over its entire domain) if for everyopen subset U of R, its preimage f −1(U) is an open subset of R.


preimage

Let f : X → Y be a map between two sets, U ⊆ Y . By definition

f −1(U) := {x ∈ X | f (x) ∈ U}

I meaningful for all functions f

I not to be mistaken for inverse function f −1 : Y → X (if thatexists — denoted the same way, unfortunately!)


preimage

Let f : X → Y be a map between two sets, U ⊆ Y and V ⊆ X .The statements

V ⊆ f −1(U)

andf (V ) ⊆ U

are logically equivalent. They both express:

for all x ∈ V , f (x) ∈ U.


preimage

Let f : X → Y be a map between two sets. The preimageoperation f −1, going from subsets of Y to subsets of X , behavesvery well with respect to boolean operations:

I f −1(U ∩ V ) = f −1(U) ∩ f −1(V )

I f −1(⋂

i∈I Ui ) =⋂

i∈I f−1(Ui )

I f −1(U ∪ V ) = f −1(U) ∪ f −1(V )

I f −1(⋃

i∈I Ui ) =⋃

i∈I f−1(Ui )

I f −1(U) = f −1(U)

I f −1(U r V ) = f −1(U) r f −1(V )

I f −1(Y ) = X

I f −1(∅) = ∅


open and closed subsets of R

Definition U ⊆ R is open if for all x ∈ U there exists ε > 0 suchthat

(x − ε, x + ε) ⊆ U

Definition U ⊆ R is closed if Rr U is open.

Remark There are several alternative characterizations of closedsubsets of the reals, but this is the one that generalizes to alltopological spaces (just wait!).


open and closed subsets of R

I (a, b) is open for a, b ∈ R ∪ {±∞}I [a, b] is closed for a, b ∈ R ∪ {±∞}I (0, 1], [0, 1), Q (the rationals) are neither open nor closed

subsets of RI ∅ and R (and only these two, actually) are simultaneously

open and closed subsets of RI arbitrary union of open sets is open

I intersection of finitely(!) many open sets is open.


Homework Problem 3

We gave three definitions of what it means for a functionf : R→ R to be continuous, and proved that Definition 1 impliesDefinition 3. Prove that Definition 3 implies Definition 1.


Remark The equivalence of Definitions 1 and 2 is classical, anddescribed in many textbooks.


metric spaces

Definition A metric space consists of a set X (“points”) equippedwith a function d : X × X → R (“distance of two points”)satisfying the properties

(1) 0 6 d(x , y)

(2) 0 = d(x , y) if and only if x = y

(3) d(x , y) = d(y , x)

(4) d(x , z) 6 d(x , y) + d(y , z)


examples of metric spaces

I R with d(x , y) := |x − y | is a metric space

I R2 with the euclidean metric is a metric space

I Rn becomes a metric space when equipped with

d(〈x1, x2, . . . , xn〉, 〈y1, y2, . . . , yn〉

):=

√√√√ n∑i=1

(xi − yi )2



I Let F [0, 1] be the set of all continuous functions [0, 1]→ R.It becomes a metric space under

d(f , g) := supx∈[0,1]

|f (x)− g(x)|

I Let X be any set whatsoever, and define

d(x , y) =

{1 if x 6= y

0 if x = y

Then X becomes a metric space!



I R2 is a metric space when equipped with

d(〈x1, x2〉, 〈y1, y2〉) := |x1 − y1|+ |x2 − y2|

I R2 becomes a metric space when equipped with

d(〈x1, x2〉, 〈y1, y2〉) := max{|x1 − y1|, |x2 − y2|}


Homework Problem 4

The ordinary notion of euclidean distance in the plane correspondsto the formula

d(〈x1, x2〉, 〈y1, y2〉) =

√(x1 − y1)2 + (x2 − y2)2

Prove using algebra, not geometry that this distance functionsatisfies the triangle inequality. Concretely, prove the algebraicinequality√

(x1 − z1)2 + (x2 − z2)2 6√(x1 − y1)2 + (x2 − y2)2 +

√(y1 − z1)2 + (y2 − z2)2

When is there an equality?



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