1 1 V13+x 3 h. lim X→0 X + 4

Q: iii) True or False: Every subspace of a complete metric space is complete.

A:

Q: i) True or False: T = {0, R} U {(-o, a]: a € R} is a topology on R.

A:

Q: Let (X, Tx) and (Y, Ty) be two topological spaces, such that X = {a, b,c, d}, Tx = {6, X, {a}, {a,…

A: I

Q: Use the laws of propositional logic to prove that the following compound propositions are…

A:

Q: Q3 Give an example of a connected subspace of R2 which is not path- connected.

A:

Q: Q1. Let X = {m,n, p, q}, {ø, X, {p}, {m, n}, {p, q}, {m, n,p}, {q}} and Y = {a, b, c}, F =…

A:

Q: Let I be the set of irrational numbers with the usual topology (that is, the topology it inherits as…

A: In topology, a component of a topological space is a maximal connected subset of the space. In other…

Q: d if -1<r e if r<1 ist the open sets in the quotient topolo

A:

Q: Q7. Assume that (R,T) is the usual Euclidean topological space. N = {1,2, 3, ....} C R. Determine…

A: There is no interior point of N.

Q: Consider the equivalence relation on R defined by r ~ y if x – y € Z. Describe the quotient space…

A: In topology and related areas of mathematics, the quotient space of a topological space under a…

Q: Two companies manufacture a rubber material in- tended for use in an automotive application. The…

A: Two companies manufacture a rubber material in tended for use in an automotive application. The part…

Q: Find the limit. lim x+v-1 (x, y)- Vx+y-1

A: The given limit is of 0/0 form. In this type of limit problems, we try to remove 0 from the…

Q: 4) Let 7 be the Euclidean topology on R and 71 be the discrete topology on R. Define f: (R, 71) →…

A:

Q: X be given by J(*) 2. Let X be the set of real numbers, let U be the usual topology, and let D be…

A: (a).         f : (X, D)  to  (X, U)   is continuous. Because if we take any set open in (X, U) ,…

Q: Let X be the set of reals, and let T = {SCX:0€ X-S} U {X}. Show that T is a topology for X and find…

A:

Q: Match the homeomorphic figures. ....F Prove or disprove the following: FA and R are pathwise…

A: Given:- (a)Homeomorphic figures     △.................. F  T.................. 0∞...................…

Q: Let X be an infinite set with the countable closed topology T-{S subset of X; X-S is countable}.…

A: Given that X is an infinite set with the countable closed topology T-S subset of X; X-S is…

Q: f(x) = {x + 2 if x > 1 (2x – 1 if x < 1 then f -'( ]2,4[ ) is then f-(]2,4[) is [1,2[ and f is not…

A:

Q: Let (X, T) and (Y, T1) be topological spaces and f a mapping of X into Y. If (X, T) is a discrete…

A: For proving f is continuous enough to prove that inverse image of every open set is open.

Q: 2. Let X be the reals and define p(x, y) = 2/x - y). Show that p is a pseudo- metric equivalent to…

A: To show that the function d(x,y) = 2|x-y| is a pseudometric on X, we need to show that it satisfies…

Q: ii) True or False: The space R is homeomorphic to the closed interval [-1,1] with the usual…

A:

Q: Which of the following sets is open in (IR, Tu)? OR-(Q-Z) (1)"2n? nin?:nE N} R - [2,5) O 12, 5) U…

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Q: A. What value of x makes the statement true: 1. x-5 = 10 2. 2x = 6 3. x/8 = 3

A: Given: x-5=10, 2x=6, x8=3 For finding value of x, we simplify given expressions

Q: Prove that every finite T,- space has the discrete topology.

A: