4. Suppose that (X, dx) and (Y, dy) are metric spaces, and define distance on X × Y by Prove that (Tmyn) → (z, y) in X × Y if and only ifFm → z in X and Un → y in Y. 5. If f:X → Y, the graph of f is the set G = {(x, f(x)) : x E X} X × Y. Suppose that f is a mapping of a compact metric space X into a metric space Y. Prove that f is continuous on X if and only if its graph G is compact. Hint: Use Exercise 4, and for the - part, also Exercise 3 applied in G.

4. Suppose that (X, dx) and (Y, dy) are metric spaces, and define distance on X × Y by Prove that (Tmyn) → (z, y) in X × Y if and only ifFm → z in X and Un → y in Y. 5. If f:X → Y, the graph of f is the set G = {(x, f(x)) : x E X} X × Y. Suppose that f is a mapping of a compact metric space X into a metric space Y. Prove that f is continuous on X if and only if its graph G is compact. Hint: Use Exercise 4, and for the - part, also Exercise 3 applied in G.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 8E: If x and y are elements of an ordered integral domain D, prove the following inequalities. a....

See similar textbooks

Similar questions

On a previous homework, you proved a Bolzano-Weirstrass theorem for ℝ3 with the metric d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 - y2| + |x3 - y3|. Conclude that, with this metric, a subset of ℝ3 is sequentially compact if and only if it is closed and bounded.
Suppose that (X,d) is a metric space and R has the standard metric. Let f : X → R be a continuous function. For each r > 0, show that A={x∈X| |f(x)|<r} is open in X.
2.) Let (S, d) be a metric space and suppose that ρ : S × S → R is defined byρ(x, y) = d(x, y)1 + d(x, y)for all points x, y ∈ S. Prove that (S, ρ) is a metric space, that it is bounded and thatρ(x, y) ≤ d(x, y) for all x, y ∈ S.
Suppose E is a subset of X, where X is a metric space, p is a limit point of E, f and g are complex functions on E and the limit as x approaches p of f(x) is A and the limit as x apporaches p of g(x) is B. Prove the limit as x approaches p of (f/g)(x)=A/B if B does not equal 0.
Let (X, T) and (Y, T1) be two topological spaces and let f be a continuous mapping of X into Y. If (Y, T1) is a T1 space, then (X, T) is a T1 space?
1. Suppose E⊆X , where X is a metric space, p is a limit point of E , f and g are complex functions on E and fx=A and gx=B . Prove fgx=AB if B≠0
Show that D^2 = {(x, y) ∈ E^2: x^2+y^2 ≤ 1} ⊂ E^2 and the space containing a single point are homotopy equivalent. (E^2 represents R^2 equipped with euclidean topology)
Let (X,d) be a metric space and d*(x,y) = d(x,y)/1+d(x,y) Prove that the family of open sets with respect to the metric d is same as the family of open sets with respect to the metric d*.
Suppose (S,d) is a metric space. How can we prove that S is open?
Let (X.d) be a metric space, x ∈ X, epsilon > 0, and E = { y∈ X: d(x,y) ≤ epsilon}. Show that E is closed.
Let (X1, d1) and (X2, d2) be separable metric spaces. Prove that product X1 × X2 with metric d((x1, x2), (y1, y2)) = max{d1(x1, y1), d2(x2, y2)} is also separable space.
Let (X,d) be a metric space. For x,y in X define e(x,y)=min{1,d(x,y)}. Prove that (X,e) is also a metric space.