1) let R be a set of real number and T={Ø, R, A₁, aeR}, where A=(a,0). Show that whether T is a topology on R or not?

Q: Let T be the Euclidean topology. Let A = [0,2] and B = [0,1[, then B is clopen in (A, TA) where TA…

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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: Consider the span of the set (V₁,..., Vn,u). Suppose u is in the span of {₁,...,Vn}. Then, show that…

A: We are given that         u ∈ span {v1 , v2 , . . . . . . , vn}    => there exist scalars a1 , a2…

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: A. Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed…

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Q: Let U be the collection of all subsets GC R, having the property that to each x = G38>0 such that…

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Q: Let R (the set of real numbers) be equipped with the Euclidean topology, then A, a finite subset of…

A: Given that R is equipped with the Euclidean topology and A is a finite  subset of R.

Q: 0 IS SU b) Let f be a map. from a space X ipto a space Y and 6 is a base for Y .show that f…

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Q: 3. Let X + 0 be a topological space. a) Show that if {U: | i € I} is a (finite or infinite) open…

A: Let X≠∅ be a topological space.  Suppose Ui:i∈I is a open cover of X.  To prove dimX=supi∈IdimUi It…

Q: Let u = and v = Show that is in Span (u, v} for all x and y. y -1 5

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Q: A. Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed…

A: Given: H is the set of all points x , y in R2 such that x2+xy +3y2=3. To Show: H is a closed subset…

Q: 5. Let X =[0,2). Define T = {[0,a):0<a<2}. %3D a) Show that T is a topology on X. b) What are the…

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Q: Let Z be the set of all integers and let R be equipped with euclidean topology t prove that tr the…

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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 &gt; a, a∈R ∪ x : x ≤ b, b∈R     To…

Q: 5) Let A= (2,3) U {4} be a subset of R with the standard opology. Find Ə(A).

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Q: Question 1. Let ACX. Show that 1= {UcX:AcU}U{®} is a topology on X.

A: For a set X having certain collection of subsets represented by τ will be a topology on X if it…

Q: 5. Let X be a set and p E X. b) If 72 = {A C X : p ¢ A}U{X}is a topology on X, then determine…

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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&gt;a,a∈R∪x:x≤b,b∈R to find the topology on R which has A  as subbases

Q: Consider the set X = {a, b, c, d} with the discrete topology, construct a basis of space that has no…

A: Consider the set X=a,b,c,d with discrete topology. Consider the basis of this set as follows.…

Q: Let τs and τ be the standard topology and the countable complement topology on R, respectively.…

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Q: 1) Let X be any non-empty set and xo E X. a) Prove that T = {X}U{AC X : xo ¢ A} is a topology on X.…

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Q: Q) Let Tright on R ? {(a, o0) : a € R}U {¢, R} . Prove that Tright is toplogy

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Q: Let lx, T) be a topological Space and A, B S X- if A is open, then show that An CIIB) s CLLANB). if…

A: Given X, τ be a topological space and A, B⊆X. Suppose, A is open then An arbitrary x∈A∩clB implies…

Q: 2. Let L={x,y,z}with the topology r={L,ø,{x},{x,y},{z},{x,z}}. Verify whether L is normal or not.…

A: The given problem is related with topology. Given that L = x, y, z with the topology τ = L, ϕ, x, x,…

Q: Let T₁ and ₂ be topologies on X. Show that J₁ J₂ is a topology on X.

A: Let us consider the two topologies, T" data-mce-style="display: inline-table; font-style: normal;…

Q: 3. Let X = {a,b,c ,d, e}, f ={{a,b}.{b.c}.fc.e}.fe}}cP(X) Find the topology z on X generated by B.

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Q: a) Consider the set X = {1,2,3} with the topology T = {Ø, X,{1}, {2, 3}, {1,2, 3}}. Show that (X, t)…

A: As per our guidelines we are supposed to answer only one asked question.Kindly repost other…

Q: Use only the definition of affine dependence to show that an indexed set {v1, v} in R" is affinely…

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Q: Prove that closed balls are closed sets in the standard topology on R?.

A: Topology Question

Q: Consider the number line R with the topology t = {Ø, R, (-0, a); a E R}. Is the set A = (-3, 0]…

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Q: Find the boise for the in- topology. Find a base for the Standord tepology-

A: This is a question of topology.

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: (1) Let (X, 7) be a topological space and let A CX. Let TA= {UNA: U ET}. (a) Show that TA is a…

A: Topology: Let τ be a collection of subsets of a nonempty set X. Then τ is called a topology if this…