4. Let X and Y be metric spaces and f : X → Y be continuous. Let SC X. Show: (b) If S is compact, then f(S) = f(S).

Q: 6. (i) Let f: (X, J) → (Y, U), where (X, T) and (Y, U) are two topolog- ical spaces. Show that if…

A:

Q: Show that d given by d(x, y) = |x-ylis a metric on R by using definition 4.2.1. You may want to use…

A: we knew  that given metric d(x,y) = |x-y| is always ≥0.  now d(x,y) = 0 ⇔ |x-y| = 0…

Q: A: For each of the following maps T: V→ W, find the spaces V and W and say if they are linear or not…

A: The given map T:V→W such that Tx, y=1, y. We have to check whether the map is linear or not.

Q: Let (X, d) be a metric space, prove that for any x, y, z E X, we have |d(y, z) - d(x, z)| ≤d(x, y).

A:

Q: Let (X, dx) and (Y, dy) be metric spaces, and let A and B be disjoint closed subsets of X X Y. Show…

A: Given:Metric spaces  and Disjoint closed subsets  and Objective: To show that there exists a…

Q: 5. Let (X,r) and (Y,A) be two topological spaces. Prove that a function f:(X.r)(Y.A) is closed if…

A: Since you have posted multiple question according to company rule we are supposed to give solution…

Q: Let X =  R x R  x R = R3 and define the metric d: R 3 x R 3 to R  by d((x1, y1, z1), (x2, y2, z2)) =…

A: Consider X=R×R×R=R3 Define the metric d:R3×R3 to R  by d((x1,y1,z1),(x2,y2,z2))=x1-x2+y1-y2+z1-z2…

Q: 1.T.2 Let V be the set of all twice-differentiable functions f : R → R satisfying f"(x) = f(x) – 5.…

A: Let, f and g be two functions in V.  Then,        f"(x) = f(x) - 5       g"(x) = g(x) - 5.

Q: Function contraction theorem Let (X, d) be a complete metric space. Suppose that f: XX satisfies…

A:

Q: Question 4. Let f:X Y be a mapping from a topological space (X,7x) a topological space (Y,Ty). Prove…

A:

Q: Let d be a trivial metric on a non-empty set X and d₁: XXX → R such that x = y d₁(x,y) = { 2 if x…

A: Given:d1:X×X→R such that  d1x, y=0if x=y2if x≠y∀x , y∈X To Show: d and d1 are equivalent.

Q: Let de(x, y) := |x — y|, d₁(x,y) := for x,y E R. Show that (a) d₁ is a metric on R; (b) dẼ and d₁…

A:

Q: A For each of the following maps T: V→ W, find the spaces V and W and say if they are linear or not…

A:

Q: Let (X, d) be a metric space and EX. Prove that if E is compact, then E is bounded.

A: Let be a metric space and . A family of subsets  is called an open cover of  if each  is open and…

Q: Let S' denote the set of (x, y) E R × R such that x? + y² This space is called the circle. Define a…

A: S1={(x,y)∈ℝ2/x2+y2=1} and p:ℝ→S1 by pt=cos2πt,sin2πt. To show: p is continuous, open,closed.

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.