Middle East Technical University Spring 1994 Math 471 Elementary Topology and Geometry

Problem List

POINT SET TOPOLOGY: GENERALITIES

Topology in a set, its basis

Problem 1. (2 pts) Describe all topologies on the set {1, 2, 3}. Construct the least basises for them.

Problem 2.

(1) (2 pts) Prove that the set of all arithmetic progressions is a basis for some topology in ℕ.

(2) (6 pts) How to use this topology to prove the infinity of the number of

Primes ?

Hint: prove that if the number of Primes is finite, then the set {1} is open (?!).

Metric spaces, metric topology

Problem 3. (2 pts) Construct a metric space and two balls in it such that the ball with the smaller radius contains the ball with the bigger one and not coinsides with it.

What is the least number of points in the space with the above property ?

Problem 4. (3 pts) Prove that the following maps ℝ^{n} × ℝ^{n} → ℝ_{+} are metrics:

(1) (x, y) → max_{i=1...n} ∥x_{i} − y_{i}∥,

(2) (x, y) → ∑_{i=1}^{n} ∥x_{i} − y_{i}∥.

(3) Describe balls and spheres in the above spaces in the case n = 2.

(4) Are these metrics equivalent ?

Problem 5. (3 pts) Let ρ: X × X → ℝ be a metric.

(1) Prove that (x, y) ↦ \frac{ρ(x,y)}{1+ρ(x,y)} is also a metric.

(2) Prove that the new metric is equivalent to ρ.

Induced Topology

Problem 6. (2 pts) Describe the topology induced on ℕ from (1) ℝ with the usual topology,

(2) ℝ with Zarissky topology,

(3) ℝ with the ”arrow” topology.

Position of points in respect to a set

Problem 7. (2 pts) Prove that Fr(A) = Fr(X − A). Find a formular for Fr(A), which contains A and X − A simmetrically.

Problem 8. (3 pts) Is it true, that Int(A ∩ B) = Int A ∩ Int B; Int(A ∪ B) = Int A ∪ Int B ? Give an example when one of the above equalities fails. Find some inclusion, which is valid for all A and B.

Dense sets

Problem 9. (1 pts) Prove that the intersection A ∩ B of two open dense sets A, B ⊂ X is also dense.

Problem 10. (2 pts) Describe the sets which are dense in the spaces with the discrete and with the antidiscrete topologies. The same question for ℝ with the ”arrow” topology and with Zarissky topology.

Problem 11. (2 pts) Let f: X → Y be a continuos surjective map and A ⊂ X is dense in X. Prove that f(A) is dense in Y.