(X,d) be a metric space. Define f: X x X -> Real Numbers by f(x,y) = d(x,y) / 1+d(x,y) . show that f is a metric on X.

Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y.

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Q: Let X and Y be two compact spaces, show that X × Y is compact

A: It suffices if we show there is a collection of open sets that covers X×Y. Let U be an open cover of…

Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y. *

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Q: Prove C(x) with the supremum norm is a metric space by showing the supremum norm is a metric

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Q: Let IR have its usual topology and let Y = {x, y, z}. Define f: R → Y by: x 0 (x, f(x) y, z, Is Y…

A: Quotient topology: If X is topological space and A is a setand if p:X→A is surjective map, then…

Q: If X =R and d(x.y)30, then X metric space not metric semi metric

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Q: Let (X,d) be metric space and X is unbounded, then d (z. y) . y EX, is 1+d(r,y) di de fined as dy =…

A: Metric space

Q: Let X = [0, 1] and d be the usual metric, d(x, y) = |x- y| %3D O True O False

A: TRUE

Q: Let (X, d) be a metric space and let A, B⊆X be such that A is connected, and A∩B ≠ ∅ and A∩ (X − B)…

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Q: Consider ℝ3 with the metric d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 - y2| + |x3 - y3|. i.…

A: Given metric is          d((x1, x2, x3), (y1, y2, y3)) = |x1 - y1| + |x2 - y2| + |x3 - y3|.   For…

Q: Let (X,d) be a metric space. For x,y ∈ X, let D(x, y) = √d(x, y). Prove that D is a metric on X.…

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Q: Let (X, d) be a metric space. Show that the function f: X X x X defined by f(x) = (x, x) is…

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Q: (v) Prove that if the space (X,T) has the fixed-point property and (Y, T1) is a space homeomorphic…

A: Given: v) To prove the given statement as follows,

Q: Consider IR as a metric space.with metric dux, y) = 1x-y1. Show that the open interval C0, 1) is an…

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Q: (b) Let (M, d) be a metric space. Define a function e: MXM-R by e(x, y) = min {1,d(x, y)}; vx,y E M.…

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Q: Let (X,d) be a metric space. Prove that the function d' :X x X →R given by d (x, y)= min {1,d(x, y)}…

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Q: A. Let H be the set of all points (x, y) in R? such that x2 + 3y2 = 12. Show that H is a closed…

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Q: Let V = {(x, y): x 20} with standard operation on R². Is V a vector %3D space?

A: Our objective is to conclude V is a vector space or not.

Q: A. Let H be the set of all points (x, y) in R2 such that x2 + 3y2 = 12. Show that H is a closed…

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Q: Let (X, d) be metric and PXIR space, dosy) I+ dexy) a de fined by qusN pusy) = ニ show that p is…

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Q: Let X and Y be compact spaces. Then X × Y is compact.

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Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y. * O…

A: Given - Let X, T and Y, T1 be two topological spaces and let f be a continuous mapping of X into Y.…

Q: Let (X, T) and (Y, T1) be two topological spaces and let f be a continuous mapping of X into Y. If…

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Q: * Let (X, d) and (Y, e) be metric spaces. Show that the function f: X xY X defined by f((r, y)) = x…

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Q: Find all limit points of a set A = {(2+,2 – 1)|n e N} on the Euclidean plane with the usual metric.…

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Q: Give the definition of a separable metric space (S, d). Show that the metric space (R, d), where…

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Q: metric

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Q: Define on R, the metric d defined by d(x, y) = |1+1리 1+ |y| Then (R, d) is not complete. True False

A: Given

Q: If X =R and d(x,y)= 1, then X not metric semi metric metric space

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Q: Let (X, t) is topological space, ACX, (X,d) is a metric space. Prove that: f: (X,d) -> (IR, Ts),…

A: Let ( X, d ) be a metric space and R with the standard topology. Then, we define a function  f(x) =…

Q: Let (X, T) and (Y, T1) be topological spaces and f a mapping of X into Y. If (X, T) is a discrete…

A: For proving f is continuous enough to prove that inverse image of every open set is open.

Q: Let (X, d) be a metric space. Define dˆ : X × X → R, by: ˆ
d(x, y) = min{1, d(x, y)}. (a) Prove that…

A: To verify that d^ satisfies the axioms of a distance function

Q: 3 Let (Y, p) be a metric space. Give an example metric d on Y such that every finction f:(Y, d)→(Y,…

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Q: Let f function from (R,d) to (R,d1) defined as d(x) = x, where d is the usual metric and .di is the…

A: Metric space function

Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y. O If f…

A: Given: (X,T) & (Y,T1) are two topological spaces & f is a continuous mapping of X into Y.…

Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y. * If…

A: Consider the given information: Let (X, T) and (Y, T1) be two topological spaces and let f be a…

Q: If X is a metric space and Y C X, then Y is a metric space.

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Q: Show that (C(X),|| · ||∞,x) is a metric space.

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Q: Let X, Y be normed linear spaces. On X x Y define || (x,y) |l, = || × |lx + || y lly and + || y ily…

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Q: the usual metric space defined by  d(x,y)= x-y prove the four propertis of metric space

A: Applying four properties to this metric

Q: For normal linear spaces X, Y, prove that BL (X, Y) is complete if Y is complete.

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Q: Prove the parallelogram law on an inner product space V; that is, showthatllx + Yll2 + llx - Yll2 =…

A: Prove the parallelogram law on an inner product space V; that is,  Show that…

Q: Let R have its usual topology and let Y = {x, y, z}. Define f: R → Y by: %3D x 0 (х, f(x) {y, (z,…

A: Quotient topology: If X is a topological space and A is set and if p:X→A is a surjective map, then…

Q: the euclidean space R³ is a separable metric space. true or false?

A: A subset M of a metric space X is said to be dense in X if M=X.X is said to be separable if it has a…

Q: Let X = R², x, y EX where x = (x₁, x2), y = (y₁.Y2) with the metric function (4 pts) (x, y) =…

A: This is a question of topology and metric space.

Q: Give the definition of a separable metric space (S, d). Show that the metric space (IR, d), where…

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Q: Show that the real number R with d(x,y)=|x-2y| are not a metric space

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Q: Let f: X, Y be a continuous surjection between metric spaces. If X is compact then A. X is complete…

A: Since X is compact , so it is complete. A option is true.

Q: Let f: (N,d) (N, d), where d is the usual metric and di is the discrete metric, defined f as f (x)…

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