(b) Let X=(2,3,5,8) with indiscrete topology and Y={a,b,c} with discrete, let f be a constant map from a space X into a space Y show that whether f is: (1) continuous map or not. (2) open map or not. (3) closed map or not. (

Q: (1) Let X be an infinite set and set r to be the set given by T = {0} U {U C X : U° finite}. (a)…

A: τ={ϕ}∪{U⊂X :Uc is finite }  (a) Xc=ϕ, which is finite . so X∈τ let A,B∈τ Then Ac,Bcare finite…

Q: Let X = (a, b, c, d) and TX = (Ø, X, (b), {b.c), {a,b,d}) be a topology 3, 4) and TY = (0, Y, (3),…

A:

Q: (a) Let X and Y be Banach spaces, T: X→ Y be a bounded linear map. Show that (i) and (ii) are…

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Q: A: For each of the following maps T: V→ W, find the spaces V and W and say if they are linear or not…

A: The given map T:V→W such that Tx, y=1, y. We have to check whether the map is linear or not.

Q: 3: Show that if T : R³ →R is a non-zero linear map, then it must be given by T = ax + by + cz for…

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Q: Bolzano-Weirstrass theorem for ℝ3 with the metric d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 -…

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Q: Q. Assume that (R,T) is the usual Euclidean topological space. Let (1,6),71.0) be the relative…

A: (a) is correct

Q: Thm. Let v be an inner product Space then and HyE V жує ||≤|12||· 1191/ further equility is hold iff…

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Q: Thm. Let v be an V be an inner product Space V and then жує | 13 11211. 11111 further equility is…

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Q: Let T V W be a linear map. : Show that Range T = Span{Tv₁,..., Suppose V Tvn}. = Span{V₁,..., Vn}.

A: Given : T : V→W be a linear map and  V = Spanv1, v2, ... , vn. To show : RangeT = SpanTv1, Tv2, ...…

Q: 2.2 Base for a Topology 39 Let X be the space of reals with the topology T = {X, [0, 1], 0} (Example…

A: The given topology is T={X,[0,1],∅}. The given sets are A=[-1,1], B=[2,3].   The derived set is the…

Q: A completely regular space is also T1-space 1 True 2 False

A:

Q: Q4. Let X = {m, n, p, q}, T = {4, X, {p}, {m, n}, {m, n, p}} and Y = {a, b, c, d}, F = {ø, X, {d},…

A: There is a slight error in the question. X cannot be an element of F. Instead of X, it will be Y.…

Q: 4) Let T be the Euclidean topology on R and T1 be the discrete topology on R. Define f : (R, 71) →…

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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: Let X be a disconnected space, that is X = UUV, where U and V are nonempty open subsets of X and…

A:

Q: 12) Check whether i) T: R³ ii) the following maps are open. R², x = (X1, X2, X3)→ (X1, X3) T: R³ R³,…

A:

Q: Let T = {0, 1, 2, 3} and the relation R is defined on T as follows: R = {(0, 1), (1, 2), (2, 2)}.…

A: The symmetric closure, R(s) is calculated using the formula, R(s)=R∪R-1 , where, R-1=b,a: a,b∈R .…

Q: (a) For the following subsets in an indicated metrie space detennine the interior, boundary points…

A: See the attachment.

Q: Q4. Let (X,T) be a topological space where X = {m,n, q, r} and T = {4, X, {q}, {r}, {q,r}, {m, q,…

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Q: Construct examples to show that if X and Y are topological spaces with G acting on them such that…

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Q: 4) Let 7 be the Euclidean topology on R and 71 be the discrete topology on R. Define f: (R, 71) →…

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Q: Let (X, t) be a topological space and A, B EX, then 1- (X- A)° CX-Ā 2- (X-A)° 2X-Ā 3- (X- A)° = X-A…

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Q: Show that R³ is spanned by S = {(1,1,1), (1, –1, 0), (0, 1, – 1)}.

A:

Q: (iii) Let X be a set with at least two points. Prove that the discrete space (X, T) and the…

A: Given : X be a set with atleast two points. To prove : Discrete space and indiscrete space ie. (X,τ)…

Q: Q # 3(b) Let (G, H) be a disconnection of a topological space (X, 7), ant let AC X. If A is…

A: Before going to this proof, we prove the below statement which in turn proves the above result.…

Q: Theorem 2.3. A set U is open in a topological space (X,J) if and only if for every point x€ U, there…

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Q: 2. Let f A B be a mapping. Prove that f is bijective if and only if f(A\C) = B\ƒ(C) for all CCA.

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Q: Thm. Let V be an inner product Space and ждек then | 1 ≤ 11all. 1171/ further equility is hold iff…

A: Using the properties of inner product And  aā=|a|2 , for any complex scalar  ||cx||=|c| ||x||, for…

Q: Let f:X Y is a function and let (B: it e A) be an indexed family of Subset of Y. Show that a)(Ua B)…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: f: Z→ No, x→ 2x, -2x - 1, when falls x ≥ 0 falls x < 0 مید ماس

A:

Q: (ii) A topological space (X, T) is said to have the fixed point property if every continuous mapping…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Theorem 2.15. Let (X,T) be a topological space, and let U be an open set and A a closed subset of X.…

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Q: (a) Let X be a topological space. Let q: X→ A be a quotient map and let p: A → B be a surjection…

A: