11.5 Problems

Problem 1. Which of the following subsets of ℝ are open, closed, or neither?

(a) ℤ (b) ℚ (c) ∩_n=1^∞(−,) (d) ∪_n=1^∞[0,1 −]

Problem 2. Let be an arbitrary set. Show that there exists a sequence such that

(An identical statement is true for as well, which can be shown in the same way.)

Problem 3. Let D(x) be the Dirichlet function, defined by

Give a mathematically rigorous proof that the limit lim_x→x0D(x) does not exist for any x₀ ∈ℝ.

Problem 4. Let X be an arbitrary set, and let τ ⊆P(X) be a collection of its subsets. (Recall that P(A) denotes all subsets of A.) The structure (X,τ) is called a topological space, if

1. ∅∈τ and X ∈τ,
2. For any collection of sets in τ, the union is also in τ,
3. For any finite collection of sets in τ, the intersection is also in τ.

The sets in τ are called open sets. Show that...