Let (X,d) be the Cartesian product of the two metric spaces (X1,d1) and (X2,d2). a) show...

Let (X,d) be the Cartesian product of the two metric spaces (X_1,d₁) and (X₂_,d₂).

a) show that a sequence {(x_n¹,x_n²)} in X is Cauchy sequence in X if and only if {x_n¹} is a Cauchy sequence in X₁ and {x_n²} is a Cauchy in X₂_.

b) show that X is complete if and only if both X₁ and X₂ are complete.

Expert Solution

Colby Messinger answered 4 months ago

(Connected Spaces) (a) Let <X, d> be a metric space and E ⊆ X. Show that...

(Connected Spaces) (a) Let <X, d> be a metric space and E ⊆ X. Show that E is connected iff for all p, q ∈ E, there is a connected A ⊆ E with p, q ∈ E. b) Prove that every line segment between two points in R^k is connected, that is Ep,q = {tp + (1 − t)q | t ∈ [0, 1]} for any p not equal to q in R^k. C). Prove that every convex subset...

1.- let(X1, τ1) and (X2, τ2) are two compact topological spaces. Prove that their topological product...

1.- let(X1, τ1) and (X2, τ2) are two compact topological spaces. Prove that their topological product is also compact. 2.- Let f: X - → Y be a continuous transformation, where X is compact and Y is Hausdorff. Show that if f is bijective then f is a homeomorphism.

(a) Let <X, d> be a metric space and E ⊆ X. Show that E is...

(a) Let <X, d> be a metric space and E ⊆ X. Show that E is connected iff for all p, q ∈ E, there is a connected A ⊆ E with p, q ∈ E. b) Prove that every line segment between two points in R^k is connected, that is Ep,q = {tp + (1 − t)q | t ∈ [0, 1]} for any p not equal to q in R^k. C). Prove that every convex subset of R^k...

Let T(x1, x2) = (-x1 + 3x2, x1 - x2) be a transformation. a) Show that...

Let T(x1, x2) = (-x1 + 3x2, x1 - x2) be a transformation. a) Show that T is invertible. b)Find T inverse.

Let (X, d) be a metric space. Prove that every metric space (X, d) is homeomorphic...

Let (X, d) be a metric space. Prove that every metric space (X, d) is homeomorphic to a metric space (Y, dY ) of diameter at most 1.

Problem 16.8 Let X and Y be compact metric spaces and let f: X → Y...

Problem 16.8 Let X and Y be compact metric spaces and let f: X → Y be a continuous onto map with the property that f-1[{y}] is connected for every y∈Y. Show that ifY is connected then so isX.

Suppose (X, dX) and (Y, dY ) are metric spaces. Define d : (X ×Y )×(X...

Suppose (X, dX) and (Y, dY ) are metric spaces. Define d : (X ×Y )×(X × Y ) → R by d((x, y),(a, b)) = dX(x, a) + dY (y, b). Prove d is a metric on X × Y .

1(i) Show, if (X, d) is a metric space, then d∗ : X × X →...

1(i) Show, if (X, d) is a metric space, then d∗ : X × X → [0,∞) defined by d∗(x, y) = d(x, y) /1 + d(x, y) is a metric on X. Feel free to use the fact: if a, b are nonnegative real numbers and a ≤ b, then a/1+a ≤ b/1+b . 1(ii) Suppose A ⊂ B ⊂ R. Show, if A is not empty and B is bounded below, then both inf(A) and inf(B) exist and...

Let x0< x1< x2. Show that there is a unique polynomial P(x) of degree at most...

Let x0< x1< x2. Show that there is a unique polynomial P(x) of degree at most 3 such that P(xj) =f(xj) j= 0,1,2, and P′(x1) =f′(x1) Give an explicit formula for P(x). maybe this is a Hint using the Hermit Polynomial: P(x) = a0 +a1(x-x0)+a2(x-x0)^2+a3(x-x0)^2(x-x1)

Let (X,d) be a metric space. The graph of f : X → R is the...

Let (X,d) be a metric space. The graph of f : X → R is the set {(x, y) E X X Rly = f(x)}. If X is connected and f is continuous, prove that the graph of f is also connected.

Question