# 2019.1Topology

 Schedule: 246T12 [13h00–14h40] Duration: 13-May-2019 – 29-Jun-2019 Classroom: B203 Study group: Telegram, link on SIGAA Contact: thanos@imd.ufrn.br Office: A225 Office hours: send email to book (first try to discuss your question on the study group)

## Contents

Calculus review. Metric spaces review. General topology: topological spaces, compactness, separation, connectedness.

## Prerequisites

This course demands a good understanding of—what I consider to be—the important topics of FMC1–FMC2 (but see below if you haven't passed FMC2 yet). To enroll, a student should be comfortable with (most of):

• common mathematical language and notation;
• writing and reading mathematical proofs in natural language (English and Portuguese);
• elementary notions of mathematical logic: syntax and basic semantics of FOL;
• very good understanding of sets, functions, relations; [12,13,15]
• elements of order theory: posets; lub/glb (sup/inf); wosets (well ordered sets); product orders; etc.; 
• limits of sequences of real numbers; 
• continuity of real functions;
• definitions by recursion and proofs by (nested) induction(s), at the very least on the natural numbers; 
• algebraic structures; group theory in particular; [17,18,19]
• infinite sets: cardinal and ordinal numbers; countable and uncountable sets; 
• Russell's paradox; and ZF should at least ring a bell; 
• etc.!

Numbers [in brackets] refer to chapters of fmcbook.

Especially about Chapter 11 (this is mandatory studying!) you can also watch the corresponding lectures of FMC2, 2018.2 on YouTube (the first three lectures on that playlist).

### Students who have not passed FMC2 yet

During the first two thirds (unidades) of FMC2 I cover most of the common prerequisites mentioned above, so if you are taking FMC2 on this semester, you should still be able to enroll in this topics course as well. (But do heed the warning above.)

### Level

WARNING! This will be a fast-paced and quite demanding course on the student's part. In the 2+ months that we have before the lectures start I will give pointers and help enrolled students brush up possibly rusty prerequisites via the study group.

## Bibliography

(Heard of libgen.io?)

### Principal

NO TEXT WHATSOEVER. We'll work with an «honor system» in which all participating students promise not to consult any related sources during the execution of this course.

### Auxiliar

The following are sources that can be helpful to study auxiliar subjects and should only be consulted for that.

### Post-course study

The following are suggestions to study after the end of this course.

• Simmons: Introduction to topology and modern analysis
• Munkres: Topology
• Kelley: General Topology
• Willard: General Topology
• Jänich: Topology
• Gelbaum & Olmsted: Counterexamples in Analysis
• Steen & Seebach: Counterexamples in Topology
• Vickers: Topology via Logic

## Exams

Written exam (date: 26/06/2019)

## Participation points

 Bianca 3 Emerson 5 Fernando Igor Gabriel Giordano 7 JP 7 Joel Felipe 4 Josenaldo 10 Lucas 5 Natália 7

## Homework

### 13/03/2019

1. Chapter 1 of [Simmons]: solve all its problems/exercises.

### 01/04/2019

1. Chapter 1 of [Munkres]: solve all its problems/exercises.

### 13/05/2019

1. Theorem (uniqueness of suprema): if a subset A of a poset X has a supremum, then it is unique.
Also (thx Jo): supA∈A ⇒ supA = maxA.
Also: any subset A of a poset X has at most one max; and if it does, maxA = supA.
2. Prove (write all the details) that the alternating sequence 0,1,0,1,0,... does not have a limit.
3. Theorem (uniqueness of limits): {aₙ}ₙ→L & {aₙ}ₙ→L' ⇒ L=L'. Thanks to this theorem we can use the notation «lim aₙ = L» which involves an equality.
4. Theorem (archimedean property of reals): If x and y are positive reals, then there is some natural number n, such that nx > y.
5. Theorem: A monotone, bounded sequence of real numbers converges (has a limit).
6. Theorem: every convergent sequence of real numbers is bounded.
7. We write infØ=+∞ and supØ=-∞. Convince a friend that this makes sense.
8. In the case of real numbers, is the following equivalent to the definition of supA?:
s = supA ⇔ (i) s is an upper bound of A & (ii) (∀ε>0)(∃a∈A)[ a>s-ε ]
9. Nested Interval Theorem: If {Iₙ}ₙ is a sequence of closed, bounded, nonempty intervals of reals that is a ⊇-chain, then ⋂ₙIₙ≠Ø. In addition, if length(Iₙ)→0, then ⋂ₙIₙ is a singleton. Reminder: «{Iₙ}ₙ ⊇-chain» means «I₀⊇I₁⊇I₂⊇I₃⊇…»
10. Exercise: Let f : [a,b] → ℝ be continuous and suppose that f(x)=0 for all rational x. Show that f(x)=0 for all x.
11. Exercise: Let f : ℝ → ℝ be continuous. (i) If f(0)>0, then f(x)>0 for all x in some open interval (-a,a). (ii) If f(x)≥0 for every rat x, show that f(x)≥0 for all real x. (iii) Will (ii) be valid if we replace «≥0» by «>0»?
12. Exercise: (i) Give an example of a bijective f : [0,1] → [0,1] that is not monotone. (ii) Can you find a monotone, injective, but not surjective? (iii) Can you find a monotone, surjective, but not injective?
13. Theorem (continuous functions respect limits): (guess the statement of this theorem by its name, and prove it).

### 24/05/2019

1. Q: Cauchy ⇒ totally bounded?
2. Q: totally bounded ⇒ bounded?
3. Q: bounded ⇒ totally bounded?

## Log

### 13/05/2019: Algebra review; calculus review

• semigroups, monoids, groups, abelians
• rings, integral domains, fields
• ordered fields
• the rationals are dense
• upper and lower bounds
• supremum and infimum (sup/inf; lub/glb; join/meet)
• axiom (law) of completeness
• complete ordered fields
• uniqueness (up to isomorphism) of complete ordered fields
• constructions of: nats, ints, rats
• construction of reals via Dedekind cuts
• limits of sequences
• a 2-player game corresponding to the definition
• continuity of function at a point
• a 2-player game corresponding to the definition
• continuity of a function

### 17/05/2019: Metric spaces

• D: distance
• D: metric space
• D: discrete space
• D: (open) balls
• D: neighborhood of a point
• D: open sets
• Θ: X and Ø are open
• Θ: binary intersection of opens is open
• Θ: arbitrary union of opens is open

### 22/05/2019: Metric spaces; Topology

• Metric spaces
• D: open set
• D: (open) balls are open sets
• D: interior of a set: int(A)
• D: limit point of a set
• D: closed set
• D: closure of a set: cl(A)
• Θ: equivalent definitions of all of the above
• D: clopen sets
• D: diam(A)
• D: d(x,A)
• D: d(A,B)
• D: derived set A'
• D: bounded set; bounded space
• D: vector space
• D: normed vector space
• D: metric induced by norm
• D: Cauchy sequence
• Q: Cauchy ⇒ convergent?
• Topology
• 1Θ1: union of two closeds is closed BiJP Lu Jo Ga JF
• 1Θ2: intersection of two opens is open BiJP Gi Lu Jo Ga JF
• 1Θ3: complement of an open is closed BiJP Gi Jo JF
• 1Θ4: complement of a closed is open BiJP Gi Jo Ga JF
• 1Θ5: each open is the union of a collection of nbhds BiJP Jo Ga JF
• 1Q1: is a singleton closed? BiJPNa Gi Jo JF
• 1Q2: is the union of a collection of closeds closed? JP JF
• 1Q3: is the union of a collection of opens open? Jo JF
• 1Q4: is the union of a finite collection of closeds closed? What about opens? BiJP Jo JF
• 1Q5: can you find a countable nbhd system for the usual topology of reals? JP Jo Lu JF Jo JP
• 1Θ6: X and Ø are clopen (both open and closed) BiJP Gi Jo Lu JF
• 1Θ7: the intersection of any collection of closeds is closed BiJP Jo JF

### 24/05/2019: Metric spaces; Topology

• Metric spaces
• D: totally bounded; ε-covering
• Θ: convergent ⇒ Cauchy
• Θ: Cauchy ⇒ bounded
• Topology
• 2Θ1: closures are closed Bi Jo JP JF
• 2Θ2: complement of interior is closure of complement JP Bi EmJF Jo
• 2Q1: p limit/interior of inter ⇒ p limit/interior of each? Bi EmJF Jo JP
• 2Q2: p limit/interior of union ⇒ p limit/interior of either? EmJF Jo JP
• 2Q3: opensubset ⇒ open? EmJF Jo JP Bi

### 27/05/2019: Topology

• D: basis of a topology
• 3Θ1: Every basis of a topological space X is a nbhd system for X, whose topology is the same as the one of X JP Bi Jo
• 3Θ2: Two nbhd systems 𝒩₁, 𝒩₂ generate the same topology iff for every point p in X: (1) for every p-hood N of 𝒩₁ there is a p-hood of 𝒩₂ contained in N (2) for every p-hood N of 𝒩₂ there is a p-hood of 𝒩₁ contained in N JP Bi Jo

### 29/05/2019: Topology

• 4Θ1: The (topological) product of two topological spaces is a topological space Bi JP Jo
• 4Q1: Is the (cartesian) product of the topologies, the topology of the (topological) product JP Bi Jo
• 4Q2: What is the product of Space 3 with itself? JP Jo Em
• 4Q3: (Y;𝒩Y) ≤ (X;𝒩)? JP Jo Bi Gi

### 31/05/2019: Metric spaces; Topology

• Metric spaces
• totally bounded implies bounded
• bounded does not imply totally bounded
• Topology
• E: a space that is not T₀
• E: a space that is T₀ but not T₁
• 5Θ1: Hausdorff ⇒ all finite sets are closed Jo Lu JP Em
• 5Q1: X Hausdorff & Y ≤ X ⇒ Y Hausdorff? JP Em Bi Jo Gi
• 5Q2: Which of the Tᵢ's is equivalent to «all finite sets are closed»? JP Lu
• 5Q3: Which of the Tᵢ's are hereditary for i=0,1,2,3 Gi JP Bi
• 5Q3': Is T₄ hereditary? Gi BiJoGi Gi
• 5Q4: Which of the Spaces 1–10 of Bing are Hausdorff? Jo

### 03/06/2019: Metric spaces; Topology

• Metric spaces
• Completeness: Cauchy ⇒ convergent
• Compactness 1: totally bounded + complete
• Compactness 2: every open cover has a finite subcover
• totally bounded ⇒ every sequence has a Cauchy subsequence
• Topology
• D: (open) cover
• D: compact
• 6Θ1: Every compact subset of a Hausdorff space is closed BiJo
• 6Q1: Is every closed subset of a Hausdorff space compact? Jo JP Em
• 6Q2: Is compactness a hereditary property of topological spaces? Jo JP
• 6Q3: Which of the Spaces 1–9 of BSL are compact? BiJo
• 6Q3': Is Space 10 of BSL compact?

### 05/06/2019: Topology

• Metric spaces
• Is the closure of the open ε-ball at x, the same as the closed ε-ball at x?
• Topology
• 7Q1: Are there T₁ spaces that are not Hausdorff? JP
• Countable ordinals
• The first uncountable ordinal Ω
• Cofinite sets
• Hereditary sets

### 07/06/2019: Metric spaces; Topology

• Metric spaces
• totally bounded ⇐ every cover has finite subcover
• complete ⇐ every collection that fips has nonempty intersection
• boundary of a set
• everywhere dense
• nowhere dense
• uniform continuity
• Topology
• D: countably compact
• 8Q1: is lp-compactness hereditary? JP
• 8Θ1: every compact is lp-compact JP
• D: fip
• 8Θ2: a topological space is compact iff it fips JP Jo
• 8Θ3: a Hausdorff space is lp-compact iff every countable open cover has a finite subcover JP
• 8Θ4: a Housdorff space is lp-compact if any countable collection of closeds fips JP
• 8Q2: is every lp-compact set compact?
• 8Q3: In 8Θ3 can we replace "Hausdorff" by "T₁"? By "T₀"?
• 8Q4: Which spaces on BSL are lp-compact?
• 8Θ5: Every closed subset of a compact space is compact JP
• 8Θ6: Every closed subset of a lp-compact space is lp-compact JP

### 10/06/2019: Cantor; Metric spaces; Topology

• Cantor
• Cantor sets
• Cantor's broom / Knaster–Kuratowski fan / Cantor's leaky tent / Cantor's teepee
• Geometric representation of the points of [0,1] in n-base system
• Cantor'sSecond diagonal argument
• Cantor's nested sets theorem
• Metric spaces
• Complete iff Cantor iff Bolzano–Weierstrass: Complete ⇒ Cantor's Nested Sets
• Topology
• 9Q1: is there a finite T₁ space that is not Hausdorff? JP

### 12/06/2019: Metric spaces; Topology

• Metric spaces
• 10Θ1: TFAE: Complete; Cantor Nested; B–W JP
• 10Θ2: compact ⇒ every continuous is uniformly continuous
• 10Θ3: Continuity: TFAE: ε-δ continuity; sequence continuity; opens continuity
• Topology
• D: Lindelöf space
• 10Q1: Which of the BSL's 1–2 are Lindelöf? Jo
• 10Q1': Which of the BSL's 3–26 are Lindelöf?
• 10Q2: Is Lindelöfness hereditary? Jo
• 10Q3: Which of the BSL's 11–26 are Hausdorff?
• 10Q4: Which of the BSL's 11–26 are compact?

### 17/06/2019: Metric spaces; Topology

• Metric spaces
• Θ: uniformly continuous functions respect totallyboundedness
• 11Θ0: In a compact space, every continuous is uniformly continuous
• Topology
• 11Θ1: E is dense iff every non-empty open contains a point of E JP
• 11Θ2: Space 1 is second-countable Jo
• 11Θ3: the topological product of two second-countable spaces, is second-countable JP
• 11Θ5: second-countable ⇒ first-countable Lu
• 11Θ6: second-countable ⇒ Lindelöf Lu Jo
• 11Θ7: second-countable ⇒ separable Jo
• 11Q1: Which of the BSL's are first-countable? Jo
• 11Q2: Which of the BSL's are separable? Jo
• 11Q3: Which of the BSL's are second-countable? Jo

### 19/06/2019: Topology

• 12Θ1: In a first-countable Hausdorff a point p is a limit point of A iff there exists a squence of distinct points that converges to p. JP
• 12Θ2: TFAE for continuous maps JP
• 12Θ3: Continuous maps respect sequences. JP
• 12Q1: Is the converse true? JP
• 12Q1: Do continuous maps respect openness? JP
• 12Q2: Do continuous maps respect Hausdorffness? JP
• 12Θ4: Composition of continuous maps is continuous. Ga JP Jo
• 12Θ5: Continuous maps respect compactness. JP
• 12Θ6: Continuity criterion for bijective maps. JP Ga

### 28/06/2019: Topology

Last update: Sun Mar 13 12:51:42 -03 2022