2019.1 Topology

Schedule:246T12 [13h00–14h40]
Duration:13-May-2019 – 29-Jun-2019
Study group: Telegram, link on SIGAA (soon)
Office hours:send email to book (first try to discuss your question on the study group)


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


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.; [22]
  • limits of sequences of real numbers; [20]
  • continuity of real functions;
  • definitions by recursion and proofs by (nested) induction(s), at the very least on the natural numbers; [11]
  • algebraic structures; group theory in particular; [17,18,19]
  • infinite sets: cardinal and ordinal numbers; countable and uncountable sets; [21]
  • Russell's paradox; and ZF should at least ring a bell; [33]
  • 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).

Please do read the links on mathematical writing and style on the Bibliography as well.

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.)


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.


(Heard of libgen.io?)


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.


The following are sources that can be helpful to study auxiliar matters 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



Probably your grade will be 50pts from homework and 50pts from a final written exam. Or something like that. It will depend on the topic selected as well.

Extra points

No extra points yet.



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


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


  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).


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

15/05/2019: No class: protest

17/05/2019: Calculus review; Metric spaces review

  • recap: complete ordered field; infimum/supremum; limits; continuity
  • distance
  • metric space
  • discrete space
  • (open) balls
  • neighborhood of a point
  • open sets
  • Theorem: X and Ø are open
  • Theorem: binary intersection of opens is open
  • Theorem: arbitrary union of opens is open

Future (fluid)

20/05/2019: Metric spaces review

22/05/2019: General Topology
















Last update: Fri May 17 17:44:20 -03 2019