Download Math 403 ASSIGNMENT #6 (due October 8) PROBLEM A (5 pt

Math 403
ASSIGNMENT #6
(due October 8)
PROBLEM A
(5 pt)
Assume that a topological space X has a countable base B = {U1 , U2 , ...}. Prove
that this space is separable.
Remark. In general, the converse statement is false: There are separable topological spaces that do not possess a countable base. But in metric spaces separability
does imply the existence of a countable base: R is a simple example.
PROBLEM B
(5 pt)
Recall that Q, the set of rational numbers, is dense in R and is countable (so R is
separable). Prove that the stronger property holds: R possesses a countable base.
Hint. Consider the family of all open intervals with rational end-points. Prove
that it is a countable base.
PROBLEM C
(2 pt)
Is the function ρ(x, y) = sin2 (x − y) a metric in R?
PROBLEM D
(3 pt)
PROBLEM E
(5 pt)
Rb
Let f be a non-negative continuous function on R. Prove that a f (x)dx = 0 iff
f (x) = 0 for all x ∈ [a, b].
Hint. If f is continuous and f (x0 ) = c then ∃ ε > 0 such that f (x) > 2c ∀x ∈
(x0 − ε, x0 + ε). This is a well-known fact from Real Analysis; you don’t have to prove
it.
.
Traditionally, C[a, b] denotes the (linear) space of all continuous functions on
Rb
[a, b] ∈ R. Prove that the function ρ1 (f, g) = a |f (x) − g(x)|dx is a metric on C[a, b].
Remark. The metric space (C[a, b], ρ1 ) is often denoted C1P
[a, b] or C 1 [a, b]. Note
that the metric ρ1 on C[a, b] resembles the metric ρ1 (x, y) = dn=1 |xn − yn | on Rd :
instead of summation over the coordinates we use integration over x ∈ [a, b]. Of
course, you are eager to study the analogs of the two other metrics ρ2 and ρmax on
C[a, b]. Sorry, I am out of space for these problems in this HWK. Next time!
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