# Q1- Show that G is transitive on X if and only if forEvery x∈X wr have G ·x =X Q2- Given n>1 in N ,let X=ZNn be the set of all sequences in the ring Zn. Show that the following correspondence defines an action of the additive group Z on X:m · (ai)i = (m + ai mod n)i  for every m ∈ Z and every sequence (ai)i ∈ X. That is, m · (ai)i is thesequence in Zn whose i-th term is given by m+ai mod n,the class of m + ai modulo n. Show as well that this action is transitive but it is neither effective nor free.1   Q3- Prove the statements in these examples (The discrete topology). τ = ℘(X). This is the topology where every subset of X is an open set. When X is endowed with this topology, we refer to X as a discrete space.AND (p-adic topology on Z). Let p be a prime number andn a nonnegative integer. For every integer a define Van={b∈Z:a≡b mod pn}.The family B={Van :a,n∈Z, n≥0}is a base for a topology on Z known as the p-adic topology and whose elements are all the unions of elements of B. Q4- Define dA : X → R bydA(x) = inf{d(x,y) : y ∈ A}. Prove that dA is bounded and uniformly continuous. Moreover, showthat for all x,y ∈ X,|dA(x) − dA(y)| ≤ d(x, y). The function dA measures how close is the point x from A. Now, LetAε ={x∈X :dA(x)<ε}. We refer to Aε as the ε-neighborhood of A.For every finite partition γ of X and every ε > 0, let γ(ε) = ? Aε × Aε.A∈γ Q5- Let (X, d) be a metric space. Define dˆ : X × X → R, by:ˆ d(x, y) = min{1, d(x, y)}.(a) Prove that dˆ is a bounded metric on X. (b) Use part (a) to prove that for ε > 0 there exists a bounded metric dˆ on X such that for allˆ x,y∈X we have d(x,y)<1⇒d(x,y)<ε.

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Q1- Show that G is transitive on X if and only if for

Every x∈X wr have G ·x =X

Q2- Given n>1 in N ,let X=ZNn be the set of all sequences in the ring Zn. Show that the following correspondence defines an action of the additive group Z on X:

m · (ai)i = (m + ai mod n)i  for every m ∈ Z and every sequence (ai)i ∈ X. That is, m · (ai)i is the

sequence in Zn whose i-th term is given by m+ai mod n,

the class of m + ai modulo n. Show as well that this action is transitive but it is neither effective nor free.1

Q3- Prove the statements in these examples (The discrete topology). τ = ℘(X). This is the topology where every subset of X is an open set. When X is endowed with this topology, we refer to X as a discrete space.

AND (p-adic topology on Z). Let p be a prime number and

n a nonnegative integer. For every integer a define Van={b∈Z:a≡b mod pn}.

The family B={Van :a,n∈Z, n≥0}is a base for a topology on Z known as the p-adic topology and whose elements are all the unions of elements of B.

Q4- Define dA : X → R by

dA(x) = inf{d(x,y) : y ∈ A}. Prove that dA is bounded and uniformly continuous. Moreover, show

that for all x,y ∈ X,

|dA(x) − dA(y)| ≤ d(x, y). The function dA measures how close is the point x from A. Now, Let

Aε ={x∈X :dA(x)<ε}. We refer to Aε as the ε-neighborhood of A.

For every finite partition γ of X and every ε > 0, let γ(ε) = ? Aε × Aε.

A∈γ

Q5- Let (X, d) be a metric space. Define dˆ : X × X → R, by:

ˆ d(x, y) = min{1, d(x, y)}.

(a) Prove that dˆ is a bounded metric on X. (b) Use part (a) to prove that for ε > 0 there exists a bounded metric dˆ on X such that for all

ˆ x,y∈X we have d(x,y)<1⇒d(x,y)<ε.

check_circle

Step 1

As per rules, (only) the first three questions are answered.

Step 2

Q1) We say a group G acts on a set X if for every g in G, we have map  T(g): X-> X satisying the properites shown.

Step 3

The idea is that every element of G acts as a permutation of the set X. Next recall the defintion of when a group action is t...

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