Let 7 be the Euclidean topology on R and 71 be the discrete topology on R. Define f : (IR, 71) → (R, 7) by f(x) = r. Determine whether f is contimuous 4) ( mapping, open mapping, or closed mapping.

Q: let x be a discrete topology Find ent (A) blit int (A) jet AC X and

A: Given that A⊂X where X is a discrete topology. To find: (i) ext (A).             (ii) b(A).…

Q: * Let R be with the indiscrete topology. If A = {1,3, 5, 7, . }, then A' = A O N O RO

A: This question is related to topology, we will use the definition of indiscrete topology to solve it.

Q: Let T be the Euclidean topology. Let A = [0,1] and B =]0.5,1] then B is clopen in (A,T) where T is…

A: YOUR ANSWER IS True.

Q: Let X be any set with the discrete topology. Show that X is normal.

A:

Q: We define the included point topology by Tp={UCR;U=Ø or pEU}. Let A = [3,5[, then A is dense in R if

A: In order for a set to be dense, it is required to be equipped in Tp and also as in the set 5 is not…

Q: 3- Let X = {a, b} write a large Topology on X. 4- If A CX, find A' for (X.I).

A: (3) It is given that,  X = a, b. Then we have to write a large Topology on X. (4) It is given that,…

Q: Let T be the finite closed topology on a finite set X, then * O Tis the Euclidean topology O Tis the…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Let R be equipped with the Euclidean topology T and let Y =]10,20[. We denote by Ty the induced…

A:

Q: Let A be any set with the discrete topology. Show that A is normal.

A:

Q: Let x be O and all subsels f X whose complements Prove that (X, z) infimite set and let z consist of…

A: Consider, I=A⊂X : Ac is countable ∪ϕSince,Xc=ϕ is countable, X∈I Also given, ϕ∈I Assume, Aii∈N∈I…

Q: 4) Let T be the Euclidean topology on R and T1 be the discrete topology on R. Define f : (R, 71) →…

A:

Q: * Let R be with the discrete topology. If A = {1, 3,5, 7, .}, then Aº A O N O RO

A:

Q: Let X be an infinite set. (a) Show that is a topology on X, called the finite co со

A:

Q: Consider a set X = {a,b} as a topological space with a discrete topology. Let J be an uncountable…

A: We will use basic knowledge of results and definitions of general topology.   We claim that Z is NOT…

Q: We define the included point topology by Tp={ UcR;U=ø or peU}. Let A = [3,5[, then A is dense in R…

A: In order to be dense in R, R has to be equipped in the Tp. Also as in the interval, 5 is not…

Q: Let f : X → Y be a function where X is a nonempty space and Y is a topological space with a given…

A:

Q: (d) Consider the set [1, √5] C [-1, √10]. Explain f is not continuous by (i) connectedness, (ii)…

A:

Q: The set S = {T, (U) | U is open in X} U {T, (V ) | V is open in Y } is a subbasis for the product…

A: The given theorem is related with the topic product topology. Given that S = π1-1U | U is open in X…

Q: Let R have its usual topology and let X = {a,b, c}. Define f : R –→ X by а, х

A: Given, ℝ have its usual topology and let X=a, b, c. Define f: ℝ→X   by…

Q: Consider the set A=(x:x>a, a E R}U(x:xsb, bE R). Find the topology on R which has A as subbases.

A: Given :               Given a set                   A = x : x > a, a∈R ∪ x : x ≤ b, b∈R     To…

Q: Suppose p: X → Y is a surjective map from a topological space X to a set Y. Verify that the quotient…

A:

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

A:

Q: Let R be equipped with the Euclidean topology T and let Y =[10,20]. We denote by Ty the induced…

A: See the detailed solution below.

Q: Consider the set A=(x:x>a, a ER)U(x:xsb, bER). Find the topology on R which has A as subbases.

A: given a set  A=x:x>a,a∈R∪x:x≤b,b∈R to find the topology on R which has A  as subbases

Q: Let T be the Euclidean topology on R and T1 be the discrete topology on R. Define f : (IR, 71) → (R,…

A:

Q: Let R be equipped with the Euclidean topology T and let Y =[10,20]. We denote by Ty the induced…

A:

Q: Let (X, T) topological space, where X = {a,b,c,d} and T = {X. Ø, {a}, {a,b}, {a,b,c}} Find a) Basic…

A: Given that X, T topological space, where X=a, b, c, d and T=X, ϕ, a, a, b, a, b, c.

Q: Let (X, T) be a topological Space and ACX- Develop the relation between Fr ( Fr(A)) and Fr(A).

A:

Q: Let R be equipped with the standard topology. Define q : R? → R, q((x1, x2) : Jx1, if x1 2 x2 |X2,…

A: Question: The function q:ℝ2→ℝ defined by qx1,x2=x1, if x1≥x2x2, if x1<x2 where ℝ2 and ℝ are…

Q: Let f.CRot,)> (R,4i) be aring homomorphism, onto function Then e

A:

Q: Let R be equipped with the Euclidean topology T and let Y =]10,2O[. We denote by Ty the induced…

A:

Q: Let T be the co-finite topology on R and d0(x,y)=|x-y|. Prove that the function f:(R,T) to (R,d0)…

A: Given that T be the co-finite topology on ℝ and d0x,y=x-y To prove that the function f:ℝ,T→ℝ,d0…

Q: Let R be equipped with the Euclidean topology T and let Y =]10,20[. We denote by Ty the induced…

A:

Q: Let A CX with the discrete topology. Prove that ô(A) = Ø.

A:

Q: 5) Let 7 be the euclidean topology on R and 7' be the finite-closed topology on R. Define f : (R,…

A:

Q: Let X be a topological space with the property that, for every topological space Y, every function f…

A: The space X with the discrete topology means every singleton of space is open.

Q: 3) let X be discrete topology and Ac X, find A ?

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: 4) Let 7 be the Euclidean topology on R and T1 be the discrete topology on R. Define f : (R, 71) →…

A:

Q: Show that the dictionary order topology on the set R × R is the same as the product topology ℝ_d × ℝ…

A:

Q: f is said to be an open mapping iff the image by f of any open set is open. Let TI be the lower…

A:

Q: Let R have its usual topology and let X = {a, b, c}. Define f : R → X by x 0 а, f(x) = с, Is X with…

A:

Q: let T be the finite closed topology on any set X.Every subset of (X,T) is connected true or false?

A: let T be the finite closed topology on any set X.Every subset of (X,T) is connected true or false?

Q: Let R be equipped with the Euclidean topology T and let Y =[10,20]. We denote by Ty the induced…

A: Let R be with Euclidean topology

Q: Let X = R, and let T consist of 0, R, and all intervals of the form (a, b) where a, b e R and a < b.…

A: A topological space is an ordered pair {\displaystyle (X,\tau ),} where {\displaystyle X} is a set…

Q: Let X and Y be normed linear spaces and F : X → Y be linear. Prove that F is continuous if and only…

A: Let X and Y be matric space and let f: x→y be a uniformly continuous function. If ( xn)n∈N is a…

Q: Let R have its usual topology and let Y = {x, y, z}. Define f: R → Y by: %3D x 0 (х, f(x) {y, (z,…

A: Quotient topology: If X is a topological space and A is set and if p:X→A is a surjective map, then…

Q: Let R equipped with the Euclidean topology. Let Y = [1,5] and denote by Ty the induced topology on Y…

A: Definition :Let (X, T) be a topological space with topology T. If Y is a subset of X, then the…

Q: Let R be equipped with the standard topology. Define q : R? → R, x1, if x1 2 x2 X2, if x1 < x2…

A: See the attachment

Q: Let (X, Tx), (Y, Ty) represents topological spaces such that X = {a,b,c,d}, Tx = {X. Ø, {a}, {a, b},…

A: CONTINUOUS FUNCTION: If the inverse image of every open subset of Y is open in X, then the function…