Solve Exercise 2.51 and Exercise 2.53 in detail please.Only attempt the question when you know the right answer. Definition 2.50. Let X and Y be sets. The product of X and Y, denoted X x Y, is theset of ordered pairs given byX × Y := {(x, y) | x € X ^ y E Y}.Exercise 2.51. Let A, C C X and B, D C Y.(1) Show that A × B c X × Y.(2) Prove or disprove: (X \ A) × (Y \ B) = (X × Y)\(A× B). (If disproved, what doeshold?)(3) Show that (A × B) N (C × D) = (An C) × (BN D).(4) Can we replace n by U in the above statement? (If not, what does hold?)Definition 2.52. Let (X, Tx) and (Y, Ty) be spaces. The product topology on X × Y is thetopology generated by the basisB := {U × V c X × Y | U € Tx ^ V E TY}.Exercise 2.53. Show that the product topology is well-defined, that is, B in Definition 2.52is a basis for a topology.Problem 2.54. Give an example to show that the basis for the product topology on X x Yis just a basis, and not generally a topology.Theorem 2.55. Let X and Y be spaces, A с Х, В с Ү, аnd give X x Y the producttopology. The Ах В — Ах В.

Name: Solve Exercise 2.51 and Exercise 2.53 in detail please.Only attempt th
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Description: Solve Exercise 2.51 and Exercise 2.53 in detail please.Only attempt the question when you know the right answer. Definition 2.50. Let X and Y be sets. The product of X and Y, denoted X x Y, is theset of ordered pairs given byX × Y := {(x, y) | x € X

Answered: Exercise 2.51. Let A, C C X and B, D…

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Solve Exercise 2.51 and Exercise 2.53 in detail please.
Only attempt the question when you know the right answer.

Definition 2.50. Let X and Y be sets. The product of X and Y, denoted X x Y, is the
set of ordered pairs given by
X × Y := {(x, y) | x € X ^ y E Y}.
Exercise 2.51. Let A, C C X and B, D C Y.
(1) Show that A × B c X × Y.
(2) Prove or disprove: (X \ A) × (Y \ B) = (X × Y)\(A× B). (If disproved, what does
hold?)
(3) Show that (A × B) N (C × D) = (An C) × (BN D).
(4) Can we replace n by U in the above statement? (If not, what does hold?)
Definition 2.52. Let (X, Tx) and (Y, Ty) be spaces. The product topology on X × Y is the
topology generated by the basis
B := {U × V c X × Y | U € Tx ^ V E TY}.
Exercise 2.53. Show that the product topology is well-defined, that is, B in Definition 2.52
is a basis for a topology.
Problem 2.54. Give an example to show that the basis for the product topology on X x Y
is just a basis, and not generally a topology.
Theorem 2.55. Let X and Y be spaces, A с Х, В с Ү, аnd give X x Y the product
topology. The Ах В — Ах В.

Expert Solution

Step 1

Note: As per bartleby instruction when more then one question is given only one has to be answered. Please upload other separately.

(1) The aim is to show that $A \times B \subset X \times Y$

Given that, $A, C \subset X, B, D \subset Y$ .

Now, it is known that,

$A \times B = \{(a, b) | a \in A, b \in B\}$

It is given that, $A \subset X, B \subset Y$ .

So, $A \times B \subset X \times Y$ .

(2) The objective is to show that, $(X \ A) \times (Y \ B) = (X \times Y) \ (A \times B)$

This can be written as,

$\begin{array}{rcl} X \ A \times (Y \ B) & = & \{(x, y) | x \in X a n d x \notin A, y \in Y a n d y \notin B\} \\ = & \{(x, y) | x, y \in X, Y a n d x, y \notin A, B\} \end{array}$

Since $x \notin A, y \notin B$ imply that, $(x, y) \notin (A \times B)$ .

So, $(X \ A) \times (Y \ B) = (X \times Y) \ (A \times B)$ .

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