For any subset A of a topological space X, we define the boundary of A to be aA = AnX\A. a) Show that int(A) and aA are disjoint and that A = int(A) U ðA b) Show that JA=Ø if and only if A is open.

Q: For any subset A of a topological space X, we define the boundary of A to be: JA = AnX\A. a) Show…

A:

Q: Prove that the open ball B(a,r) is a subset of the interior of the closed ball C(a,r)

A: Given that Ba,r is an open ball of radius r and centered at a. Then a∈Ba,r                ...... (1)…

Q: Let Y be an infinite subset of a compact set X c R. Prove Y' # Ø.

A: Y' denotes the derived set of Y i.e. the set of all limit points of Y. We just need to show that any…

Q: For the following sets in the metric space (IR, d), determine A' and A. i). (1,2] ii). [0, 1) i).…

A:

Q: Exercise 2. Assume that (X, T) is a topological space. a) If A, B are subsets of X with AC B, show…

A:

Q: Let fex) and g(x) be two (Piece - wi'se) comtinuous functions on Intervál caib]. Then if t) ¿gCx)…

A:

Q: ► Exercise 3.7. Suppose X is a topological space and UC SCX. (a) Show that the closure of U in S is…

A:

Q: ► Exercise 4.4. Prove that a topological space X is disconnected if and only there exists a…

A:

Q: E. Let H be the set of all points (x, y) in R² such that x² + 3y² = 12. Show that H is a closed…

A: We have to find H is closed  subset bounded.

Q: Exercise 2. Let X be a topological space. Endow Xx X with the product topology. Consider the map f:…

A:

Q: Exercise 8. Assume A, B are subsets of the topological space X. a) If AC B, show Å C B. b) Show that…

A: a)

Q: Let $X$ be a set and assume that p E X. Let T, be the collection (Ø, X, and all subsets of X that…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Let (X, d) be a metric space. The diameter of a non-empty subset M of X is defined as 8(M)= sup d(x,…

A:

Q: If V=R\power{3}, and W\index{1} is the xy plane and  let W\index{2} is yz-plane:…

A: Let (a,b,c) be any element of R3, then (a,b,c)=(x,y,0)+(0,y,z)=(x,2y,z) ⇒a=x, b=2y, c=z ⇒a,b,c∈R ⇒…