Question

In: Advanced Math

Let X be a subset of R^n. Prove that the following are equivalent: 1) X is...

Let X be a subset of R^n. Prove that the following are equivalent:

1) X is open in R^n with the Euclidean metric d(x,y) = sqrt((x1 - y1)^2+(x2 - y2)^2+...+(xn - yn)^2)

2) X is open in R^n with the taxicab metric d(x,y)= |x1 - y1|+|x2 - y2|+...+|xn - yn|

3) X is open in R^n with the square metric d(x,y)= max{|x1 - y1|,|x2 - y2|,...,|xn -y n|}

(This can be proved by showing the 1 implies 2 implies 3)

(TOPOLOGY)

Solutions

Expert Solution

Colby Messinger answered 2 years ago
Next > < Previous

Related Solutions

1. Let A be an inductive subset of R. Prove that {1} ∪ {x + 1...
1. Let A be an inductive subset of R. Prove that {1} ∪ {x + 1 | x ∈ A} is inductive. 2. (a) Let n ∈ N(Natural number) and suppose that k 2 < n < (k + 1)2 for some k ∈ N. Prove that n does not have a square root in N. (b) Let c ∈ R \ {0}. Prove that if c has a square root in Z, then c has a square root in...
Prove the following: Let f(x) be a polynomial in R[x] of positive degree n. 1. The...
Prove the following: Let f(x) be a polynomial in R[x] of positive degree n. 1. The polynomial f(x) factors in R[x] as the product of polynomials of degree 1 or 2. 2. The polynomial f(x) has n roots in C (counting multiplicity). In particular, there are non-negative integers r and s satisfying r+2s = n such that f(x) has r real roots and s pairs of non-real conjugate complex numbers as roots. 3. The polynomial f(x) factors in C[x] as...
Let x, y ∈ R. Prove the following: (a) 0 < 1 (b) For all n...
Let x, y ∈ R. Prove the following: (a) 0 < 1 (b) For all n ∈ N, if 0 < x < y, then x^n < y^n. (c) |x · y| = |x| · |y|
Let φ : R −→ R be a continuous function and X a subset of R...
Let φ : R −→ R be a continuous function and X a subset of R with closure X' such that φ(x) = 1 for any x ∈ X. Prove that φ(x) = 1 for all x ∈ X.'  
Use induction to prove Let f(x) be a polynomial of degree n in Pn(R). Prove that...
Use induction to prove Let f(x) be a polynomial of degree n in Pn(R). Prove that for any g(x)∈Pn(R) there exist scalars c0, c1, ...., cn such that g(x)=c0f(x)+c1f′(x)+c2f′′(x)+⋯+cnf(n)(x), where f(n)(x)denotes the nth derivative of f(x).
hzhshs assume that X is normed linear space let A is a subset of X prove...
hzhshs assume that X is normed linear space let A is a subset of X prove that: if A is compact then A is closed and bounded
. Let x, y ∈ R \ {0}. Prove that if x < x^(−1) < y...
. Let x, y ∈ R \ {0}. Prove that if x < x^(−1) < y < y^(−1) then x < −1.
Suppose that f(x)=x^n+a_(n-1) x^(n-1)+⋯+a_0∈Z[x]. If r is rational and x-r divides f(x), prove that r is...
Suppose that f(x)=x^n+a_(n-1) x^(n-1)+⋯+a_0∈Z[x]. If r is rational and x-r divides f(x), prove that r is an integer.
Let A be an infinite set and let B ⊆ A be a subset. Prove: (a)...
Let A be an infinite set and let B ⊆ A be a subset. Prove: (a) Assume A has a denumerable subset, show that A is equivalent to a proper subset of A. (b) Show that if A is denumerable and B is infinite then B is equivalent to A.
Let R be a commutative ring with unity. Prove that f(x) is R[x] is a unit...
Let R be a commutative ring with unity. Prove that f(x) is R[x] is a unit in R[x] iff f(x)=a is of degree 0 and is a unit in R.
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT