Math 140B, Lecture A, Spring 2008: Compactness & Path-Connectedness in Metric Spaces - Pro | Assignments Calculus

Mathematics 140B, Lecture Section A, Spring 2008, Homework #3 (due Apr 21, end of lecture)

Everywhere below, Rkand its subsets are assumed furnished with the metric d(x, y) = |x−y|, unless noted

otherwise.

1. (Ross 13.12a). Let (S, d) be a metric space, Fa compact set in (S, d), and Ea closed subset of F. Prove that

Eis compact.

2. (Ross 13.12b). Prove that finite unions of compact sets are compact.

3. Let (S, d) be a metric space. A continuous function f: [a, b]→Sis called a path. (Here a<b.) A pair of

points pand qin Sare said to be path-connectable if there exists a path f: [a, b]→Swith f(a) = pand

f(b) = q. A set Ein (S, d) is said to be path-connected if every pair of points in Eis path-connectable by a

path f: [a, b]→E.

(a) Find all the subsets of Rthat are both compact and path-connected.

(b) Find a path that connects the points p= (3,7) and q= (−1,0) in R2.

4. Let (S, d) and (S∗, d∗) be metric spaces, f:S→S∗a continuous function, and Eis a subset of S.

Prove each of the following.

(a) If Eis path-connected, then f(E) is path-connected.

(b) If f: [a, b]→Ris continuous, then there exist points xand xin [a, b] such that

i. f(x) = infx∈[a,b]f(x) and f(x) = supx∈[a,b]f(x)

ii. For every point y∈[f(x), f (x)] there exists a point x∈[a, b] such that f(x) = y.

[Hint: Is [a, b] compact? Path-connected?] Notice that this is a stronger version of the Intermediate Value

Theorem from Math 2A.

the base of the mountain. The next day, the monk spends from 6am to 6pm to come down and reach the

base. Is there a time of day on which the monk is at the same height on day 1 and on day 2? Prove your

finding.

5. Let (S, d) be a metric space. A function f:S→Sis called a contraction mapping on (S, d) (or just a

contraction on (S, d)) if there exists a constant C < 1 such that

d(f(p), f (q)) < Cd(p, q) for all p, q in S

Intuitively this means that fdecreases the distance between a pair of points by a factor >1/C.

(a) Prove that every contraction mapping is continuous.

(b) Prove that if fis a contraction mapping on a complete metric space (S,d), then there exists one, and only

one, point xin Ssuch that f(x) = x. (This point is called a fixed point of f.The above theorem can

therefore be worded as follows. Every contraction on a complete metric space has a unique fixed point.)

[Hint: Arbitrarily pick a point x1∈S, and define the sequence (xk)∞

k=1 by xk+1 =f(xk). Is this sequence

Cauchy?]

computational procedure for finding that solution approximately.

6. Ross, #21.9.

7. Ross, #21.10.

8. Ross, #21.11.

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Mathematics 140B, Lecture Section A, Spring 2008, Homework #3 (due Apr 21, end of lecture) Everywhere below, Rk^ and its subsets are assumed furnished with the metric d(x, y) = |x − y|, unless noted otherwise.

(Ross 13.12a). Let (S, d) be a metric space, F a compact set in (S, d), and E a closed subset of F. Prove that E is compact.
(Ross 13.12b). Prove that finite unions of compact sets are compact.
Let (S, d) be a metric space. A continuous function f : [a, b] → S is called a path. (Here a < b.) A pair of points p and q in S are said to be path-connectable if there exists a path f : [a, b] → S with f (a) = p and f (b) = q. A set E in (S, d) is said to be path-connected if every pair of points in E is path-connectable by a path f : [a, b] → E.

(a) Find all the subsets of R that are both compact and path-connected. (b) Find a path that connects the points p = (3, 7) and q = (− 1 , 0) in R^2.

Let (S, d) and (S∗, d∗) be metric spaces, f : S → S∗^ a continuous function, and E is a subset of S. Prove each of the following.

(a) If E is path-connected, then f (E) is path-connected. (b) If f : [a, b] → R is continuous, then there exist points x and x in [a, b] such that i. f (x) = infx∈[a,b] f (x) and f (x) = supx∈[a,b] f (x) ii. For every point y ∈ [f (x), f (x)] there exists a point x ∈ [a, b] such that f (x) = y. [Hint: Is [a, b] compact? Path-connected?] Notice that this is a stronger version of the Intermediate Value Theorem from Math 2A. (c) A monk takes from 6am till 6pm to climb to the peak of a mountain of height M. The climb starts at the base of the mountain. The next day, the monk spends from 6am to 6pm to come down and reach the base. Is there a time of day on which the monk is at the same height on day 1 and on day 2? Prove your finding.

Let (S, d) be a metric space. A function f : S → S is called a contraction mapping on (S, d) (or just a contraction on (S, d)) if there exists a constant C < 1 such that

d(f (p), f (q)) < Cd(p, q) for all p, q in S

Intuitively this means that f decreases the distance between a pair of points by a factor > 1 /C.

(a) Prove that every contraction mapping is continuous. (b) Prove that if f is a contraction mapping on a complete metric space (S, d), then there exists one, and only one, point x in S such that f (x) = x. (This point is called a fixed point of f. The above theorem can therefore be worded as follows. Every contraction on a complete metric space has a unique fixed point.) [Hint: Arbitrarily pick a point x 1 ∈ S, and define the sequence (xk)∞ k=1 by xk+1 = f (xk). Is this sequence Cauchy?] (c) Prove that the equation sin(x) − x = 0 has a unique solution in the open interval ]0, π/2[. Indicate a computational procedure for finding that solution approximately.

Ross, #21.9.
Ross, #21.10.
Ross, #21.11.

Math 140B, Lecture A, Spring 2008: Compactness & Path-Connectedness in Metric Spaces - Pro, Assignments of Calculus

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