Let (X, sx) (Y, sy) be two partially ordered sets. se+ Xx Y is defined as The relation son the (a, b) s (C, =) as x c and bsyd for each, a) show. that (X x 9, S) is a partlally ordered set. b)Draw the praph of the set (Xxy,s) with the of the par tially ordered sets X = {a, b, c 3 and y = { die, f? as follows. %3D (x,

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A: a x,y : -1<x<1 , y=0b x,y : x and y are integersc x.y : x+y =1d x,y : x+y<1e x,y : x=0 or…

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A: Let X be any infinite set with co-finite topology. Definition:  Let X be any infinite set. The…

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Q: 21. Let A and B be two on-empty bounded sets of positive embers and let С%3D {ху: х€ А and y € B}.…

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Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is F open?

A: A set is open if and only if it contains all of its limit points. Also recall that the union of open…

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A: Answer is mentioned below

Q: Let X, Y be two sets. Prove that X × Y C P(P(X UY )).

A: It is given that X,Y  be two sets. To show that X×Y⊆PPX∪Y Firstly, we define the set X×Y as, X×Y=x,y…

Q: Let X, Y be two sets. Prove that X × Y C P(P(X U Y )).

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A: Everything You have done was correct except the second last step . So, I m doing from there . Since…

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

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A: We will solve three subparts of the given problem as per the answering guidelines. If you want…

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A: Solution: Before going into the problem we must know the Theoem named Heine-Borel Theorem.…

Q: O Show that if Si and S2 are convex sets in R™X", then so is t mxn S = {(x, y1 + y2) | x € R", y1,…

A: Convex set: , If a and b are points in a vector space S. Then the vector space is called convex set…

Q: Let X, Y be two sets. Prove that X × YC P(P(X U Y )).

A:

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A: The given relation:R=x,y| x is a proper subset of y.

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A: Given,relation R defined in the set A of all triangles as R = {(T1, T2) :T1 is similar to T2}.

Q: 1. Prove that for all sets X, Y , if XU Y = Xn Y, then X = Y. %3D

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A: A = {c, d, f} and B = {w, x}, B x A = { (w,c), (w,d), (w,f), (x,c), (x,d), (x,f) }

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…

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A:

Q: Let X be an infinite set endowed with the finite complement topology T, then (X,T) is compact. * O…

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Q: let U and y be bounded sets. if u er i tre M+ (V) ≤M₂ (V) and M^ (U) SM³ (V) 1

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Q: Show whether the sets are connected or not. 1. {z € C | 1< ]z| < 2} 2. D(0, 1) U D(2i, }),

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

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Q: Prove that the boundary points of an open interval (a, b) CR are the points a and b.

A: A point a∈ℝ is said to be boundary point of a set A⊂ℝ, if Ba,r∩A≠∅ and Ba,r∩Ac≠∅ for every…

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Q: Suppose that R (the set of real numbers) is equipped by TO (the Euclidean topology), T1 (the…

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

Q: Let X, Y be two sets. Prove that X x YC P(P(X UY )).

A:

Q: Let f : (X,d) - (Y, d1) becontinuous fumction, then for any closed set G in X, f(G) is not necessary…

A: False

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Q: Let S be the subset of the ℝ2 given by all pairs (x, y) so that x2 + y2 = 2. Show that S is a closed…

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Q: f A∩B≠∅ and A∩Bc≠∅ then show that A∩X\B̅≠∅ and A∩B̊≠∅

A:

Q: Let the non-empty set S = {x - yv3:x,y ER}, show that S is closed'under multiplication then find the…

A: We have to show that the given set S=x-y3:x,y∈R is closed under multiplication and also find the…

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Q: let E be a set that baunded. Prove that Inf E = Sup -E) where R, XE E y

A: To prove for a buonded set E inf(E) = -sup(-E) , where -E = {-x, x belongs to E} , we first prove…

Q: Suppose that R (the set of real numbers) is equipped by TO (the Euclidean topology), T1 (the…

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

Q: If (X,, T,), (X, T2) are two topological spaces, and A,, A, are open sets in X,and X, respectively,…

A: