Question

1. a) State the definition of a topological space. b) Prove that every metric space is...

1. a) State the definition of a topological space.

b) Prove that every metric space is a topological space.

c) Is the converse of part (b) true? That is, is every topological space a metric space? Justify your answer.

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Prove that if T is a discrete topological space (i.e., every subset of T is open)...
Prove that if T is a discrete topological space (i.e., every subset of T is open) and ∼ is any equivalence relation on T , then the quotient space T / ∼ is also discrete topological space. [Hint: It is a very short and straightforward proof.]
A topological space is totally disconnected if the connected components are all singletons. Prove that any...
A topological space is totally disconnected if the connected components are all singletons. Prove that any countable metric space is totally disconnected.
Prove that the Gromov-Hausdorff distance between subsets X and Y of some metric space Z is...
Prove that the Gromov-Hausdorff distance between subsets X and Y of some metric space Z is 0 if and only if X = Y question related to Topological data analysis 2
1.- Prove the intermediate value theorem: let (X, τ) be a connected topological space, f: X...
1.- Prove the intermediate value theorem: let (X, τ) be a connected topological space, f: X - → Y a continuous transformation and x1, x2 ∈ X with a1 = f (x1), a2 = f (x2) ( a1 different a2). Then for all c∈ (a1, a2) there is x∈ such that f (x) = c. 2.- Let f: X - → Y be a continuous and suprajective transformation. Show that if X is connected, then Y too.
1) a). Prove or Disprove: "Every random variable has cumulative distribution". Justify b). If statement in...
1) a). Prove or Disprove: "Every random variable has cumulative distribution". Justify b). If statement in problem (a) is true, does the existence of cumulative distributions function of a random variable imply the existence of probability density function? Justify.
Let X be a topological space with topology T = P(X). Prove that X is finite...
Let X be a topological space with topology T = P(X). Prove that X is finite if and only if X is compact. (Note: You may assume you proved that if ∣X∣ = n, then ∣P(X)∣ = 2 n in homework 2, problem 2 and simply reference this. Hint: Ô⇒ follows from the fact that if X is finite, T is also finite (why?). Therefore every open cover is already finite. For the reverse direction, consider the contrapositive. Suppose X...
4. Is complete metric space always compact? Answer the question by considering the real line R....
4. Is complete metric space always compact? Answer the question by considering the real line R. Justify your argument in detail.
These problems concern the discrete metric. You can assume that the underlying space is R. (a)...
These problems concern the discrete metric. You can assume that the underlying space is R. (a) What does a convergent sequence look like in the discrete metric? (b) Show that the discrete metric yields a counterexample to the claim that every bounded sequence has a convergent subsequence. (c) What does an open set look like in the discrete metric? A closed set? (d) What does a (sequentially or topologically) compact set look like in the discrete metric? (e) Show that...
Let A and B be independent events of some sample space. Using the definition of independence...
Let A and B be independent events of some sample space. Using the definition of independence P(AB) = P(A)P(B), prove that the following events are also independent: (a) A and Bc (b) Ac and B (c) Ac and Bc
Prove that if ? is a finite dimensional vector space over C, then every ? ∈...
Prove that if ? is a finite dimensional vector space over C, then every ? ∈ ℒ(? ) has an eigenvalue. (note: haven't learned the Fundamental Theorem of Algebra yet)
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT