Q3/(a) Let d be a trivial metric on a set of X, find the induced topology on X by a metric d. (b) Let T be a discrete topology on a set of numbers, show that whether (R, T) is: (1) is To-space or not. (2) is T₁-space or not. (3) is normal space or not. (4) is a compact space or not.

Q: Q2) Let (A,T)be topological spaces; i=1,2,3. Then A, x A2 x Az is homomorphic to (A, x A2) x A3 I

A: Definition : Let X and Y be spaces, and let B be the collection of sets U × V , where U ⊆ X and V ⊆…

Q: Prove that any contraction between metric spaces has at most one fixed point

A: According to the given information, it is required to prove that any contraction between metric…

Q: 1/ Let (X₁T) and (Y₁J.) be top spaces and fix-x be a surjective and continuousmap, then y is Compact…

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Q: ) Consider the space R^2 equipped with the sunflower metric. Identify explicitly the set of points…

A: The objective of the question is to identify the set of points in R^2 that form a ball of radius 1…

Q: 11. Let (X, d) be a pseudometric space, and suppose d has the property that d(a, b) >0 whenever ab.…

A: 11) To prove that every finite subset of X is closed, we need to show that its complement is open.…

Q: 6. Show that a compact metric space X is locally compact.

A: The given problem is related with metric space. We have to show that a compact metric space X is…

Q: every metric space is normed space? give an example to show tha

A: We have to show that every metric  space  may not be a  normed  space.

Q: Let (R,d) be diserete metric space then R is not compact . True or false.??

A: Let (R,d) be diserete metric space then R is not compact . True or false.??

Q: (c) Prove that a metric space is connected iff it contains exactly two sets that are both open and…

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Q: Find two examples for equivalet norms on three dimensional real space R³. .

A: To take two vectors whose norms are the same.

Q: Define quotient space of a vector space.

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Q: Verify that the discrete metric is actually a metric.

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Q: Using the fact that A/B = An Be, show that in a metric space, bdy(A) = bdy (Aº)

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Q: Show that a set A in ℝ^ 2 is open in the Euclidean metric ⇔ it is open in the max metric.

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Q: 5) Let X be a topological space. Is there always a metric on X which gives rise to its topology?…

A: No, there does not always exists a metric in a topological space that gives rise to its topology.…

Q: if (X,d) is discrete metric space, ACX and A+D (x,d) Find the sets of A° and A (functional analysis…

A: Consider a discrete metric space X,d where d is defined as: dp,q=0,p=q1,p≠q

Q: If X is a metric space with induced topology Ƭ, then (X,Ƭ) is Hausdorff. The contrapositive of this…

A: Given : (1) (ℝ, T) with the topology induced by taxicab metric.              (2) (ℝ, T) with the…

Q: Let (X, t) be a topological space and A, B EX, then 1- (X- A)° CX-Ā 2- (X-A)° 2X-Ā 3- (X- A)° = X-A…

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Q: M is an infinite set. If x,y ∈ M, explain the relation f below            {  1 , if x = y   f(x,y)…

A: We have to solve given problem:

Q: (a) Prove that the metric space (2,11) is complete,

A: Complete Metric Spaces:- A Metric Space (X,d) is said to be complete iff every Cauchy sequence in X…

Q: 4={(-21.r):reR} II) B=(2r.-31): te R} III) C={(2r+3,-41) re R} I)

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Q: Q1) Prove or Disprove: Let L(F) be a linear space and A,B are two convex subsets of L. Then, AXB is…

A: The problem is related with convex set. Given that,                 LF  is a linear space  and A, B…

Q: (b) Suppose -|, and ||-||2 are two equivalen norms on a vector space X. If (X, ||.||,) is complete,…

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Q: 3. For a metric space (X, d), the diameter of a subset A C X is defined by 6(A) := sup d(x, y).…

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Q: Define what it means for a topological space to be completely regular,

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Q: (a) Show that every metric space is T2-space.

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Q: one example of a 4-dimensional vector space that is not R 4

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Q: Consider the empty set, i.e. X={ }. Which property of vector space is not satisfied and how?

A: A vector space is a set with a 0 element with two operations: addition and scalar multiplication…

Q: Q6: (a) In any metric space (S, d), prove that every compact subset T is complete. (b) Prove that…

A: A metric space is a set together with a metric on the set. The metric is a function that defines a…

Q: Tet fiR*-o R. be dafined by it ² tyz if X=y =O Show that f is qutinyous on IR

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Q: B. Is C a real vector space? Explain.

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Q: prove that a formation d:AxA->R is a metric on the set a if and only if (1) and (2) apply

A: Um zu zeigen, dass eine Abbildung \(d: A \times A \rightarrow \mathbb{R}\) genau dann eine Metrik…

Q: the usual metric space defined by  d(x,y)= x-y prove the four propertis of metric space

A: Applying four properties to this metric

Q: Q3:- Let ((0,4], d₁) be a metric space and the open ball in this metric B₂ (3) = (1,4]. Show that…

A: A metric space is a mathematical structure that consists of a set of elements along with a distance…