5. If X any Y are sets, then X-Y {r€X: z4Y} If X is open set and Y is closed set. Prove that (a) X-Y is open.

Q: e following sets of points is collinear 2. L(-5,13), M(-10,10), N(-20,4)

A: Here we have to prove that each of the following sets of points is collinear.  L(-5,13); M(-10,10),…

Q: What set is in another class of infinity that is bigger than R?

A:

Q: 1

A: Consider the provided question, We have to prove by using only definition of a compact set, that (0,…

Q: Thoorem: is not closed set

A:

Q: 5. Let E be a set that bounded and c > 0. Prove that sup(cE) = -c inf(-E) %3D

A: This is a problem of Real topology, metric topology.

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…

Q: Let (R,r) be topological space with usual topology on R, then the set Qof rationals is O Neither…

A: ℚ is the set of rational numbers. ℝ is the set of real numbers. So, ℚ is a subset of ℝ .        i.e.…

Q: Consider two sets in ℝ² with the usual topology. Is the closure of a path-connected set also…

A: We will use knowledge of general topology to solve both the parts of this question. For the first,…

Q: Queston2 {e-x, xe-O onlo, c∞) a) shau that q fandamented Set

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Q: (2) Show that if f has support in a compact set A CR and g has support in a compact set B C R then f…

A:

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

A:

Q: Show that if A is closed in Y and Y is closed in X, then A is closed in X.

A:

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={a, b, c, d, e} and let T={X, non zero, {a}, {a, d}, {a, e}, then (X, T) is neither Hausdorff…

A:

Q: Determine whether is a linearly independent set of vec- 3 11 tors in R3.

A: A set of vectors v1,v2,v3,…,vk in a vector space is said to be linearly independent if…

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

A:

Q: Prove that ifS is a bounded set, then inf(S) = -sup(-S) %3D

A:

Q: 21. Let A and B be two non-empty bounded sets of positive numbers and let C = {xy : x e A and y e…

A:

Q: We defined the relation between sets by A ~ B means that there there is a 1-1 correspondence ƒ : A →…

A: We define the relation ~ between sets by A~B means that there is a one-one correspondence f:A→B Show…

Q: 5. Show that if A and B are bounded subsets of R, then A U B is a bounded set. Show that sup(A U B)…

A:

Q: 4. Prove that the intersection of an arbitrary collection of compact sets in R is compact.

A: (Heine-Borel Theorem). A subset A of R is compact if and only if A is bounded and closed (with…

Q: Suppose metric space X is not path-wise connected then X is not connected. O True O False

A: Question 9 Answer: as a path connected space of employees connected but converse is not true. So I…

Q: 4.2. Let S be the open set consisting of all points z such that |2| < 1 or |z – 2| < 1. Show that S…

A: Connected set: If any two points of a given set can be joined by a continuous polygon arc such that…

Q: Let Y be an infinite subset of a compact set X ⊂ R. Prove Y' does not equal ∅.

A:

Q: let a be a nonempty compact subset of R and let B be a nonempty closed sub set of R such that A…

A: Let A be a non-empty compact subset of ℝ Let B be a non-empty closed subset of ℝ such that A∩B=∅ To…

Q: Show that if A is closed in X and B is closed in Y then A × B is closed in X × Y

A:

Q: Let X = {a, b, c, d,e} and let T = {X, ø, {a}, {a,d}, {a,e}}, then * %3D (X, T) is not Hausdorff but…

A:

Q: The continuous image of a connected set is connected. That is, if X is connected and f : X–→ Y is…

A: We prove this via contradiction. We assume that f(X) is not connected then X is not connected. And…

Q: 4. Show that if A is closed in Y and Y is closed in X, then A is closed in X. Verify it and make…

A:

Q: Express (4, 10) as a set {x : |x − a| &lt; c} for suitable a and c

A: Write the given interval in inequality form as follows: 4&lt;x&lt;10

Q: Let (X, CA) be a measurable space, f: X (-00, ws) and 1: X (-∞, ∞0) are measurable. Prove that the…

A:

Q: Use the definition of compact set to prove that the set [0,1] is compact

A: We know that a set E is compact if every open cover of E has a finite subcover. We suppose, that O…

Q: 3. ** Let X, Y be finite sets, and f is a function XY. What can you say about the cardinality if X,…

A: 3. Given X, Y are finite sets and f:X→Y is a function   To Comment about cardinality of X and Y if…

Q: Let X = {a, b, c, d,e} and let T = {X, Ø, {a}, {a,d}, {a,e}}, then * (X, T) is not Hausdorff but it…

A: Since you have asked multiple question,as per our guidelines we are supposed to answer only one…

Q: Đ A is a closed set.

A: We prove by using the result that intersection of two closed set is closed.

Q: Show that B,(x) is open set with respect to 7 for all r> 0.

A:

Q: Let X = {a, b, c, d, e} and let T ={X, Ø, {a}, {c,d}, {a,c,d}, {b,c,d,e} }, then O XT) is neither…

A: Option 1 is correct.

Q: Show that the interval [0,1] is a closed set in R

A:

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

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Q: T have to show rigorous proof of the inclus (An B) x C c (Ax C)UCBXC) 1,B.and C, Sets non-em pty of…

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Q: EX LeT X = {!,2,3,415,6} T- Ex,¢,てり,E4,しろ4), [ス456 Topology.?

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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 closed?

A:

Q: a nonempty compact set of IR. sup Ke K. be

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Q: 28

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Q: 3.1 Show that an infinite intersection of closed sets F, k 1,2, 3,..., in a metric snace (S, d) is a…

A:

Q: Is it true that (A×A)∖(B×B)=(A∖B)×(A∖B) for any two sets A, B? If it is true, prove it. Otherwise,…

A: Explanation of the answer is as follows

Q: Let A, B, C be arbitrary finite sets from the same universal set U. - (a) Is it true that A - B C (A…

A: SOLUTION-

Q: Prove that if A is a nonempty set which is bounded below by 3 then B = {x ∈ R | ∃a ∈ A, b = (1/a)}…

A: Given that A is non empty set which is bounded below by 3. Given that B=x∈R|∃a∈A,b=1a The objective…