Consider the standard topology on R, Let A=[3,7], the interior of A is Oa A O b.6 Oc. (3,7) OdR

Q: Q3.Let (R, T) is the usual Euclidean Topology. Assume that A = {1,2, 3}, B = (2,4] Determine d(AU B)…

A: Given set A={1,2,3} and B=(2,4]

Q: Given the bases B= (u1,u2} and B' = {u'1, u'2} for R, where and u'1= The transition mat rix PB-B'…

A: In this question we have to find the transition matrix from B to B'

Q: Given the bases B = {u,,u2} and B'={u'1, u'2} for R², where and u'1= The transition mat rix Pg-8'…

A: Given bases  are B=u1,u2 and B'=u1',u2'  where u1=10,u2=01 and u1'=12,u2'=11 To find the transition…

Q: , Evaluate by interpreting it in terms of areas. *P (2* – I^ + *),J

A: We have to evaluate integral by interpreting in terms of area

Q: Let E be any Euclidean sp The following properties hol ap f: EE is an isometry fof=

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Q: Question 3. Consider the topology r= {X,ø,{a},{a,c},{a,b,d}} X = (a,b,c,d). Let A= {a,d}, find A and…

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Q: For the metric space (R, d) with d the usual Euclidean metric, which sets are in the topology T…

A: Wr Know that,"Let X = (-∞, ∞) and C consists of all intervals of the form (a,b).The arbitary union…

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: If y is orthogonal to both and then y is orthogonal to every w in Span{u, v} V. O True False

A: We have to solve given problem:

Q: Which of the following is a topology on R? {UCR: U is infinite}U{U<R: U is countable} O {[a, b): a,…

A: We will use the definition of Topology.

Q: Given the bases B={u , u2} and B'={u'1, u'z} for R2, where , U2 and u'2 = The transition mat rix…

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Q: Which of the following is not a base for any topology on R? a. {(a, b]: a,b ER and a <b} O b. {[a,…

A: Base for a topology: The base for topology of any set X is collection B of subsets of X which…

Q: Q7. Assume that (R,T) is the usual Euclidean topological space. N = {1,2, 3, ....} C R. Determine…

A: There is no interior point of N.

Q: The dimension of the set W = {(2t – s, s, t) : s, t e R} of R is equal to: %3D

A: The dimension of the set W={(2t-s,s,t) : s,t ∈ R} of R3 x=2t-s y=s z=t (t,s) ∈ R   This can be…

Q: If h:(R,+) → (R*,.) is a mapping defined by h(x)=10*, VXER, then (R,+) = (R*,.).

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: 4. Consider the space (R,T), where T={R,Ø}u{AcR:[0,1)cA}. If A=(0,1), then the closure of A is: * O…

A:

Q: In the open disk z-z>r ylgn O ihi O

A: Here we have to show that the statement is true or false, the statement is given as:In the open disk…

Q: Question 3. Consider the topology 1={X,.{a),{a,c),{a,b,d}} X = (a, b,c,d). Let A = {a,d}, find A and…

A:

Q: The standard metric on R is d: R? → [0, 0) defined by d(x, y) = |x – yl. a. Show that the resulting…

A: The given standard metric on : The distance given: a. To prove the basis of standard topology on…

Q: Let X = R, and d be the diserete metrie, and A= {1,2) Show that if A is compact or not

A: Given set A={1,2}

Q: The collection {a} × (b, c) C R² | a, b, c E R} of vertical intervals in the plane is a basis for a…

A:

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: Let 7 be the Euclidean topology. Let A [0,2] and B [0,1, then Bis dopen in (AT) where 7, is the…

A:

Q: Given the bases B={u, u} and B'={u'1, u'z} for R2, where -1 Uz and u'2 The transition mat rix Pg-B…

A:

Q: LetX={(x, y)∈R×R:xy= 1}.Suppose thatR×Rhas thestandard topology, andXhas the relative topology.…

A:

Q: Which of the following is not a base for any topology on R? O {(-0,a): a E (-∞, 0]}U{(0, ∞)} O {(a,…

A: Base for a topology: Let X, τ be a topological space. A collection of open sets B is said to be base…

Q: et S be a nonempty subset of R and let k E R. Define kS {ks : s E S}. Prove if k > 0, en sup(kS) = k…

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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&lt;x2 where ℝ2 and ℝ are…

Q: Q4. Let (R?,T) is the usual Euclidean Topology. Given M = {(x, y), 1 < x² + y² < 4} C R°. Determine…

A: See the attachment

Q: Question 1. Let AcX. Show that t= {Uc X :ACU}U{ø} is a topology on X.

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

Q: If P and Q are represented by the numbers z1 and z2 such that H- then the circumcentre of AOPQ,…

A: Given P an Q are represented by the numbers z1 and z2 such that |1z1+1z2|=|1z1-1z2| We need to find…

Q: Show that the linear map (x, y)H (x+ y, x- y) from (R?, || - |1,) to (R?, I| - ||is an isometry.

A:

Q: Ir X = {a,b, c, d, e, S} and T is the discrete topology on X, which of the following statements are…

A: Given: X={a,b,c,d,e,f} τ is the discrete topology on X.   a)X∈τ By the definition of topology. The…

Q: Consider the following topology on X = {a,b, c, d, e}: T = {X,0, {a}, {c, d}, {a,c, d}, {b, c, d,…

A:

Q: Given the bases B = {u,u2} and B'= {u', u'2} for R?, where and u'= The transition mat rix Pg-g' from…

A: Linear transformation problem.....

Q: Let P = {(z, y, 2) E R° | az + by + cz = 0} where (a, b, c) # (0,0,0). (a) Prove that (a, b, c) ¢ P.…

A: Given that P=x,y,z∈ℝ3 : ax+by+cz=0 where a,b,c≠0,0,0 (a) The objective is to prove that a,b,c∉P Let,…

Q: Let 1={Ø}U{UER:NC U}, -n+3 ). A = :n EN and B = [0, ∞). Then n+1 in the topological space (X, 1) O…

A:

Q: Let T={(a, 0): aER}, then T is Borel algebra.

A:

Q: 4. Consider the space (IR,T), where T={R,Ø}u{AcR:[0,1)cA}. If A=(0,1), then the closure of A is:…

A:

Q: Give an isometry on C taking AABC to AA'B'C' for (i) A = A' = 0; B = B' = 2, C = i and C' = -i. (ii)…

A:

Q: 3. Ir X = {a,b, c, d,e, f} and T is the discrete topology on X, which of the following statements…

A:

Q: Let X = {a.b.c.d.e) Find the topology r on X generated by f-{{a.b.c).(b.c.d).fc.d}}

A: X={a,b,c,d,e}

Q: Let J={(a, ∞ ) : a€R}, then T is Borel algebra. ulgn O

A: Given a statement

Q: 2. Let T be the cofinite topology on R, and let A = (-x, 1) U (1, ), B = (1,2). Fine the boundary…

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Q: Consider R with the left ray topology. If A={2, 4, 8} the derived set of A is O a. (8, 00) Ob. A Oc.…

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Q: Consider the sets N = R? \ {(0, y) E R² | |y| > 0}, %3D L = {(x, y) E R² | |¤| = |y|}, and let di,…

A:

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 J={(a,∞): a€R}, then T is Borel algebra. صواب İhi

A: True

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