6. Let A and B be two connected subsets of X and An B + 4. Show that AUB is connected. In R*(eqquiped with the Euclidean metric), find if the union of A = {(x,y): x² + y s 1} and B = ((x,y): (x - 2)2 + y < 1} is connected or not? Justify. Let (X, d) be a compact metric space and let d, be the metric on X defined by d, (x, y) = min{ 1, d(x, y)}. x, y EX Then prove that (X, d,) is compact.

Q: Let M = {0,1} and d be the discrete metric, then d is not complete %3D O True O False

A: Discrete metric space.

Q: 3.3 Let r1 and r2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

A: 3.3 is also known as Hausdorff Property. d(x1, x2) &gt; 0 for every x1 ≠ x2   (metric space…

Q: Sketch a model that satisfies atleast 2 axioms of Fano's Geometry and Identify those axioms.

A: In this type of question, first we need to know, about Axioms of  Fano's Geometry, after this, we…

Q: Show that the closed ball Y in a metric space (X,d) is a closed set. Also show that if (X,d) is…

A: Show that the closed ball Y in a metric space (X,d) is a closed set. Also show that if (X,d) is…

Q: Prove the following theorem in neutral geometry: If there exists a constant c and a model for…

A:

Q: Consider the closed rectangle R = [a,b] × [c,d] in ℝ2 (under the Euclidean metric). i. Explain why…

A: Consider the closed rectangle R = [a, b] × [c, d] in ℝ2 (under the Euclidean metric). For (i), We…

Q: a. Show that Ø, R E T. (by definition of Topology)

A:

Q: Show that Loo is not icomorphic to a subepace of

A: Consider the spaces l∞   and   lp where p∈1, ∞. It is known that l∞ is not a separable space but, lp…

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: The Cartesian product X = X1 x X2 of two metric spaces (X1, d1) and (X2, d2) can be made into metric…

A: 4) To define a new metric on X which is different from the metrics mentioned in (1), (2), and (3).…

Q: 3) I Consider the topology = {GCR:0 ¢ Gor G = R} on R. Is open %3D %3D interval (-1, 1) compact in…

A:

Q: 3. Show that X = [0, 1] with the discrete metric is bounded but not totally bounded. %3D

A:

Q: Consider a 2-space with the following metric ds² = rdr² + ydy? (a) Calculate the nonvanishing…

A: Given,A 2-space metric: ds2=x dx2+y dy2To find: a) Connection coefficients, Γ111 and Γ222…

Q: Let X and Y be two T, spaces, then X×Y. :is

A:

Q: Let X = N and d be the usual metric, d (x, y) = |x – yl then d can induces a norm on X True O False…

A:

Q: Show that X = [0, 1] with the discrete metric is bounded but not totally bounded.

A:

Q: The set [0, 1] with the discrete metric is compact. True False

A:

Q: Consider 2-dimensional hyperbolic space with metric ds ds'+dy and y > 0. The length of a line…

A:

Q: Show that the closed interval [0, 1] and the unit circle S' = {(x, y)|x² + y² = 1} cannot be…

A: Suppose there exist a homeomorphism f:[0, ∞)-----&gt;R then restriction of f f|[0,∞)\{0} : [0,…

Q: 5. If мая abstract manifolds, PEM. f a) Define TpM and dfp, where M²₂N is smooth map beton by dfp:…

A:

Q: (b) Show that x. y < ||x|| ||y||- [Hint: If x, y # 0, let a = , 6= and use the fact that ||ax + by |…

A: Given that, Euclidean metric d on Rn is defined as d(x,y) = || x - y || for x, y in Rn Part(a): Let…

Q: With the help of exomples Show thaat on finite dimensional are equivalent. any two norms Linear…

A: Given: Any two norms on finite dimensional linear space are equivalent.

Q: The set [0, 1] with the discrete metric is compact. O True O False

A:

Q: Show that in R with the euclidean metric, any subset of R \Q with more than one element is not…

A:

Q: Prove that: 1. (R, |. |) is a complete metric space. 2. (Q.I-1) is not complete metric space.

A:

Q: Consider A = [3,7), B = (7,9] as a subset of R with the usual metric  dR (x,y) = abs(x-y). Show Ac…

A:

Q: The set R- {0} with the usual metric is connected.

A:

Q: we diveagena theosem to eveluate ty3j + XzK. dA ahose sethe boundary Of cube defined by

A:

Q: The sphere x² + y² +z² = 100 has the center (0,0,0) and a radius of length 50. Select one: True…

A:

Q: 3.4 Give the definition of a separable metric space (S, d). 3.5 Show that the metric space (R, d),…

A:

Q: Let (X, d) be a metric space and B = S(xo,r) be the open ball with centre at xo E X and radius r> 0.…

A:

Q: let M= To,2] anel d be bhe usual metric then d is not compact or Compact ?

A: A set is said to be compact if and only if it is closed and bounded.

Q: Draw the unit “circle” around the origin in the Euclidean metric, the taxicab metric, the hub…

A: The unit circle around the origin in Euclidean metric is The unit circle around the origin in…

Q: Show that every closed ball in R" is an n-dimensional manifold with Assuming the theorem on the…

A:

Q: If X = {a, b} , how many metric can be de fined on X %3D infinite O none O only one O only finite

A: only one

Q: Let (R, d) be discrete metric space, then R is not separ able O True O False

A:

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

A:

Q: Prove by sequences that the open balls in R^n with the usual (Euclidean) metric are not compact

A: In this question, we have to show that open balls in ℝn with the usual metric are not compact by the…

Q: Consider 2-dimensional hyperbolic space with metric ds = dr'+dy and y > 0. The length of a line…

A:

Q: Let (S, d) be a metric space and suppose that ρ:S×S→R is defined by ρ(x, y) =d(x, y)/(1 +d(x, y))…

A:

Q: Show that by using an example that every metric is a pesudo metric but converse is not true.

A: To show that every metric is a pesudo metric but converse is not true.

Q: d(r, y) Let (X, d) be a metric space and let p(r, y) = for all r, y E X. Show that (X, d) 1+d(x, y)…

A: According to our company policy we are supposed to answer only first question i.e. (4) 4) in this…

Q: Consider ỹ,,y2, ÿ,ER³ and that y,+y,+y;=0. For Î=span{ ỹ,,y} and Y=span{ỹ2, y, show that Î=Ỹ.

A: Given  that  y→1, y→2, y→3∈ℝ5 and that                   y→1+ y→2+ y→3=0→. and           Y^=spany→1,…

Q: Show that the set of all real numbers constitutes an incomplete metric space if we choose d(x, y) =…

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

Q: Let S be the portion of the cylinder y = 1 - x2 with x≥0; y≥0, bounded by the planes z = 2, and z =…

A: Consider the given information. The formula for the double integral is defined as.…

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

A:

Q: Prove that: 1. (R, J. |) is a complete metric space. 2. (Q, I. |) is not complete metric

A: To prove that R,. is a complete metric space.  Suppose that xn is increasing and bounded, Let…

Q: Let S be the portion of the cylinder y = 1 − x2 with x≥0; y≥0, bounded by the planes z = 2, and z =…

A: