Topology

Second Semester 2025

Thursdays 10:30am - 12:30pm 31J
Fridays 11:30am - 12:30pm 14J

Syllabus (here) (Spanish aqui)

Book:
Introduction to Topology (Dover Books on Mathematics)  Bert Mendelson (here in DJVU)

Homeworks: 

HW 1. Due Friday August 8
Problems: Cap. 2 Sec. 2 Prob: 1,2,3,7
Prove that   a.b <= |a||b| for a,b in Rn.
Consider X with the discrete metric D, and any other metric d which is not equivalent to the discrete. Consider the identity function Id: (X, D ) ->( X, d), is it continuous? Is  Id: (X, d ) ->( X, D), continuous?

HW 2. Due Thursday Agust 14
1. Show that the function x->1/x is continuous from R+ to R+.
2. Show that the sum function (x,y)-> x+y from RxR to R is continuous (use the taxi metrix on RxR).
3. Show that the multiplication function (x,y) -> xy from RxR to R is continuous (use the maximum metric on RxR)
4. Show that composition of continuous functions is continuous (show that (x,y)->x/y is continuous)

HW 3. Due Thursday August 21
1. Let f:X ->Y be a function. Show that f is continuous if and only if the function f sends converging sequences to converging sequences.
2. Let f:X ->Y be a function. Show that f is continuous if and only if the inverse image of open sets is open.
3. Problems: Chap 2. Sec. 6. Prob: 2,3

FIRST EXAM (Chap 2. Sec 1-8) 

HW4. Due Thursday August 28
1. Take a set X and consider the subset of P(X)  consisting of all subsets of X with finite complement and add the empty set. Show that this is a topology on X. Call this topology  the finite complement topology.
2. Take the reals R with the  finite complement topology. Show that the polynomials  in one variable p(x) are continuous functions from R to R in this topology. Hint: Work with the closed sets instead of the open ones. Recall that a function is continuous if the inverse image of a closed set is a closed set.
Problems: Chap. 2 Sec.7 Prob: 1,2

HW5. Due Thursday  September  4
1. Let {T_i]_{i \in A} be a set of topologies on the set X. Show that the \cap_{i \in A} T_i defines also a topology on  X.

Problems: Chap. 3 Sec. 4 Prob: 1,4,5 (Pag 102)

HW6. Due Thursday September 11
Problems: Chap. 3 Sec. 5 Prob: 5 (Pag 107)- Sec. 6 Prob. 6 (112)- Sec. 7 Prob. 4 (115)

SECOND EXAM (Chap 3. Sec 1-7)  September 11

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HW7. Due Thursday
Problems: Chap.4 Sec. 2 Prob. 4, Sec. 3 Prob. 2,3 Sec. 4 Prob. 2 Sec. 6 prob. 3

THIRD EXAM Chap. 4 Sec. 1-6 

HW8. Due Thursday
Problems: Chap. 5. Sec. 2 Prob 1,2,3,4

HW9. Due Thursday
Problems: Chap. 5. Sec. 3 Prob 3, Sec. 5. Prob. 2,4,6

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Homeworks: Almost weekly.
They must be written by hand in white paper written only using one side of it.
They must be handed is at the beginning of class.

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Spanish
Deben presentarse en HOJA BLANCA tamaño carta. Escritas por un sólo lado y en letra legible.
Las tareas estarán disponibles desde el viernes y deberán ser entregadas a más tardar el jueves de clase a las 2:30pm.
No recibiré tareas en otro tipo de hoja, ni tendré en cuenta tareas enviadas por correo electrónico.