Let U be subset of R". Fill in the blank to define when U is an open set. U is called open if z € U, U.e > 0, Je>o,

Q: et A and B be sets. Show that if AUB=U, then ACB. (Note: U is the universal set.)

A: A universal set (usually denoted by U) is a set which has elements of all the related sets, without…

Q: Let A be a set. Prove that S = 0. SEP(A)

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: ) (AO B) © (A^ B) = A UB.

A: Introduction :

Q: 4- A subset [a, b] R is compact set in: a- (R, Td), b- (R, T), c-Both (R, Tu) and (R, T₁) d- No one.…

A: As per our guidelines we are supposed to answer only first part. Kindly repost other parts in the…

Q: • Show that if D is a set and E is a closed set containing D, then E must contain ƏD, i.e., the…

A: according to our guidelines we can answer only three subparts, or first question and rest can be…

Q: Assume that a and b are real numbers such that a <b. Prove that the interval (a,b) is an open set.…

A: Given that a and b are real numbers and a&lt;b. Prove that (a,b) is an open set

Q: Let neN. Set 4, = {n,n+1,n+2,..}and 7 = {ø, A,} pen · a) Verify that T is a topology on N. b) Is…

A:

Q: (ii). Prove that a set S CR is everywhere dense iff it intersects every non-empty open sets in…

A:

Q: Let A be a compact set in R. Show that the complement A is open and contains an interval of the form…

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

Q: let x be an infinite set and let ta be a topology on x in which all infinite subsets of x are open…

A: Given that X be an infinite set and let τ be a topology on X in which all infinite subsets of X are…

Q: = U.

A:

Q: Consider R² with the usual topology, let A and B be two subsets of R2. We denote the closure of a…

A:

Q: The subject is on Elements of Set Theory:I really need help and explanation.

A: The symmetric difference of two sets are defined here as,

Q: Suppose A and B are finite sets. Prove or disprove: (aun-aun)- (AUB) = (A u B) ) + Dsjnt(A, B)

A: Consider the equation

Q: (A'n B) = A' U B'.

A: Prove that : (A ∩ B)' = A' U B'

Q: Consider S to be the universal set. Let X = (AUB) nC and Y = (AnT)U (An B). %3D Write out the…

A:

Q: Let X = {a, b, c, d}, S = {{a, b, c}, {a}, {b, c, d}}, and T be the topology on X generated by S.…

A:

Q: 5. If r = (r, (x, y, z) and ro = (xo, Yo, 20), describe precisely the set of all points (r, y, z)…

A: We plug the given values in the given equation and simplify

Q: 5) I Let R be the set of all real mumbers with the cuclidean topology. Define the set A {(r, y) E IR…

A:

Q: Let U ⊆ ℝ be an open set. Then U is the union of a countable, disjoint collection of open intervals.…

A: Given: U=∪i∈Iai, bi where I is a countable set, ai, bi∈ℝ, ai=−∞, or bi=∞ for all i∈I. To show: ax∉U

Q: Every set S has an open cover.

A: If S is a subset of  R,   Then every set S  will always have an open cover.

Q: Let X be a non empty set and p be a fixed point in X. Define T={X, AcX such that peA}. Then {p} is:…

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

Q: Let U = {x | x is a letter in the word %3D GUADALARAJA}, where this is the universe set. Also, let M…

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: find fundamental set of soleubions cwith d real va lues for

A: Using differential equations,

Q: give a Reel number r>0 and a∈ X in (X, ||.||) norm space. Determine whether B[a,r] is convex by…

A: Given that X is a normed space with norm . Let r&gt;0, be a real number. Then closed ball Ba,r is…

Q: Let A and B be sets. Prove that the intersection of A and B is is a subset of the union between A…

A: To show that the elements which are common to both A and B belong to either A or B.

Q: Q: In usual topological space (R.Tu). answer the following: 1- Is [0,1] is compact set in R? Why?…

A: Note: Hi! Thank you for the question, as per the honor code, we are allowed to answer three…

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

A: No information is given for topology on Y So i have solved only the middle question

Q: 4) LetXbe any set with the discrete topology. Let Y be any subset of X. Prove that ô( Y ) =Ø.

A:

Q: Let U be the universal set with disjoint subsets A and​ B, where n(U) = 37​, n(A) = 8​, and n(B) =…

A: first we will find the value of n(A'∩B') we know according to De Morgan's law  A'∩B' =…

Q: Let U be the universal set such that any set is a subset of U. Let A and B be sets. Use element…

A:

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

A: Given :  U and V are bounded sets with  U⊆ V. To show :  m*(U) ≤ m*V  and  m*U ≤ m*V,  where m*U and…

Q: Let X be an infinite set, then any topology on X is also infinite. Select one: O True O False

A: Note: " Since you have asked multiple question. As per our guidelines we are supposed to solve only…

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: Let A and Bbe finite sets satisfying |A|= 9, B| = 6 and |An B = 2. Calculate the following: () |AUB…

A: As per our company guidelines we are supposed to answer only first 3 sub-parts. Kindly repost other…

Q: Let X = (6,7), and d be the ususal mortric, d(r, y) = |r - yl Gire ezample of a set ACX, to be not…

A:

Q: Let'Y be the set of f bleod types B+ Hat type Connot donate blood to.

A: Here, X is the set of blood types AB- can donate blood. AB- can donate blood to AB- and AB+ Hence,…

Q: Let X be an infinite set and T be a topology on X. If every infinite subset of X is in T, then T is…

A:

Q: (c) let t be indiscrete topology on set of integers Z, show that whether (Z, t) is regular space or…

A: Given : ℤ , τ is indiscrete topological space. To show : To show whether ℤ , τ is regular , T0, T2,…

Q: Let X be an infinite set and I be a topology on X. If every infinite subset of X is in T, then Tis *…

A:

Q: Set C and the universal set U are defined as follows. U={p,q,r,x,y,z} C={p,q,x}

A: Given,        Set C and the iniversal set U are defined as follows :         U=p,q,r,x,y,z…

Q: Let T be the finite closed topology and be also the discrete topology on a non empty set X. Then…

A: X is finite

Q: let A Set bounded belou, erove that C. inf A Inf C A ン

A:

Q: Prove that a set S⊂R is everywhere dense iff it intersects every non-empty open sets in R. Prove…

A:

Q: (2.2) Let C = {a, b, c, d} and T = {0, {a, b, c} , {b, c} , {a},C} be the topology defined on C. %3D…

A:

Q: Are the sets Z x Z; ((x.y): x*y?) closed in (R , | |) ? What about the sets (0,1]U(2,3); [0, 00) in

A:

Q: Let X = {a, b, c, d,e} and let T= { X, Ø, {a}, {a,d}, {a,e} }, then * O (X,T) is Hausdorff and…

A: