Question: A set E in a metric space is called clopen if it is both open and closed. a) Prove that every metric space has at

A set E in a metric space is called clopen if it is both open and closed.
a) Prove that every metric space has at least two clopen sets.
b) Prove that a metric space is connected if and only if it contains exactly two clopen sets.

Step by Step Solution

★★★★★

3.43 Rating (166 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a By Remark 1011 and X are clopen b Suppose E is clopen and E X Then U E ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

741-M-N-A-D-I (596).docx

120 KBs Word File

Students Have Also Explored These Related Numerical Analysis Questions!

A metric space X is connected if and only if X and are the only sets that are both open and closed.

A subset S of a metric space is both open and closed if and only if it has an empty boundary.

A set A is called clopen if and only if it is both open and closed. a) Prove that every Euclidean space has at least two clopen sets. b) Prove that a proper subset E of Rn is connected if and only if...

. Answer the following questions. 1. (3 points) In Metropolis only taxicabs and privately owned automobiles are allowed to use the highway betwem the airport and downtown. The market for taxi cab...

EXAM II: PRACTICE SHEET (1) All the HW problems! (2) (a) Let (X, d) be a metric space. Prove that the closed neighborhood N r (x) = {y X : d(x, y) r} is a closed set. (b) Let (Rn , d) with the...

Kindly solve the following questions. 8. Find a limit point for the set {-1 : ne ( ). Bolzano 9. Let A = {-1 : ne N}. a) Find A in R. b) Explain why A is a compact subset of H. 10. Let X be a compact...

Convex geometry 0.1 Dehn-Sommerville relations In this section we will talk about convex polytopes. One possible denition of a convex polytope uses the notion of a half-space: A closed half-space in...

Let A, B be sets. Define: (a) the Cartesian product (A B) (b) the set of relations R between A and B (c) the identity relation A on the set A [3 marks] Suppose S, T are relations between A and B, and...

please answer 3. Observe that if (X,T) is a discrete space or an indiscrete space, then every open set is a clopen set. Find a topology T on the set X = {a,b,c,d} which is not discrete and is not...

Choose the correct description(s) of reverse transcriptase. Reverse transcriptase is a DNA polymerase that adds dNTPs to aDNA primer. Reverse transcriptase is a DNA polymerase that adds dNTPs to aDNA...

Touchie MacFeelie publishes comic books. The only inputs he needs are old jokes and cartoonists. His production function is Q = .1J1/2L3/4, where J is the number of old jokes used, L the number of...

Debt transactions are often few in number but large in dollar amount. As such, auditors are generally able to substantiate individual transactions rather easily.

Identifying tests for two samples. zdemir (2013) stated, when trying to detect and describe differences between two independent groups, by far the most common strategy is to use the arithmetic mean...

9. Let f E COO (a, b). Prove that f is analytic on (a, b) if and only if f' is analytic on (a,b).

8. (a) Prove that for n E N. (b) Show that 00 f(k)( ) f(x) = L T xk , k=O x E R. 2.9253

7. Suppose that f E COO (-00, 00) and for all a E R. Prove that f is analytic on (-00,00) and

Zambezi Corp. expects EBIT to vary with the state of the economy as follows: 560 State of economy EBIT in $ million) Boom Base 125 Bust -200 Zambezi doesn't have any long-term debt and has 95 million...

Case 1 Case 2 Case 3 Direct material used 20,000 12,300 Direct labor 11,000 Manufacturing overhead applied 12,000 Total current manufacturing costs 29,000 28,500 Beginning work in process inventory...