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 is Closed. what would happen-

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: Let (X, d) be a metric space and let A ⊆ X be a compact subset. show that, If B ⊆ X and d(A, B) =…

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Q: For any subset A of a topological space X, we define the boundary of A to be: JA = AnX\A. a) Show…

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Q: Let (R, d) be usual metric space, then Ris ot separable True False

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Q: Let (X, d) be a metric space and let A ⊆ X be complete. Show that A is closed.

A: Given that X,d be a metric space and A⊆X be complete. The objective is to show that A is closed.

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: We define the included point topology by Tp={UCR;U=Ø or pEU}. Let A = [3,5[, then A is dense in R if

A: In order for a set to be dense, it is required to be equipped in Tp and also as in the set 5 is not…

Q: If U is homeomorphic to V and U is compact, then * V is complete O V is connected V is compact V is…

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Q: If there exists open ball B (a, r) such that B (a,r) nG = Ø then a is not limit point of G True…

A: TRUE

Q: (c) If X= R, with r={ R, ¢, Q, Irr}. Show that (R, t) is a compact space.

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Q: Let x be O and all subsels f X whose complements Prove that (X, z) infimite set and let z consist of…

A: Consider, I=A⊂X : Ac is countable ∪ϕSince,Xc=ϕ is countable, X∈I Also given, ϕ∈I Assume, Aii∈N∈I…

Q: * Let R be with the discrete topology. If A = {1, 3,5, 7, .}, then Aº A O N O RO

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Q: Let A be a closed bounded subset of a metric space (X, d). Is A a compact subset? Explain why.

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Q: Let (X,d) be a metric space. For x,y in X define e(x,y)=min{1,d(x,y)}. Prove that (X,e) is also a…

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Q: Let (X,t) be a topological space, A, B C X. Then A' UB' = (A U B)', A'n B' c (A N B)' Select one: O…

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Q: Consider a set X = {a,b} as a topological space with a discrete topology. Let J be an uncountable…

A: We will use basic knowledge of results and definitions of general topology.   We claim that Z is NOT…

Q: We define the included point topology by Tp={ UcR;U=ø or peU}. Let A = [3,5[, then A is dense in R…

A: In order to be dense in R, R has to be equipped in the Tp. Also as in the interval, 5 is not…

Q: Let

A:

Q: Let A and B be any two subsets of a metric space (X, d). Then (1) A is a closed set. (2) If AC B,…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: let (x, m) be a measurable space. then ¥8:% [, o] are measural ble funetion, then ) I +9 is mea…

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

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Q: Let X be an uncountable set and let oo EX be fixed. Let TX = {U E P(X); 0$U or X- U is finite} Let Y…

A: Given: Let us consider the given X to be unaccountable and let ∞∈X be fixed.

Q: Let (X,d) be any metric space, then the only open sets and closed sets are and X O True O False

A: Given statement is false because if we take descrete metric space

Q: Let A be a non-empty subset of the metric space (X,d). show that |d(x, A)- d(y, A)| < d(x, y)For all…

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Q: Let ( Xr d be ame tric space,show that exist atopology on X induce b

A: Let (X, d) be a Metric Space We know that in a Metric Space  (1) arbitrary union of open sets is…

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

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Q: Let X = {a,b,c} prove that topological space on the set X T = {X. Ø, {b}, {a,b}} be a

A: Let X={a,b,c} be a set. To show that T={X,ϕ,{b},{a,b}} be a topological space on the set X.

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

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Q: Let (X, T ) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f ∈ C(X, M);…

A: Given as X,T be a topological space,M,d be a complete metric space and BCX,M=f∈CX,M;fX is bounded…

Q: Let X= [0, 0 ), and defined d as d(x,y)= e(xy) then verify all four condition of metric space

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Q: Let (X, T) topological space, where X = {a,b,c,d} and T = {X. Ø, {a}, {a,b}, {a,b,c}} Find a) Basic…

A: Given that X, T topological space, where X=a, b, c, d and T=X, ϕ, a, a, b, a, b, c.

Q: Let (X, T,) be the particular point topological space with respect to X. Then for any subset, find…

A: This is a question of topology.

Q: Let A and B be any two subsets of a metric space (X, d). Then (1) A is a closed set. biso (2) If…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: Show that the closed ball Y in a metsic SPace (X,d)is a closed set. Also, show that it (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: Let (C[a,b] .d) be the discrete metric space. Then Cla,b] is bounded. * False True

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

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Q: Let X be a set with more than one element. (X,t) is a topological space such tha Vxe X,{x} €T then…

A: Given a set X with more than one element.

Q: Let X be an uncountable set and let T = {U C X :U = Ø or U¢ is finite} %3D Then is topological space…

A: let X be an uncountable set and T=U⊑X : U=∅ or Uc is finite i.e cofinite topology on X. Then the…

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

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

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

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Q: If U is homeomorphic to V and U is compact, then * O Vis compact V is complete V is connected O V is…

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Q: 4. Let (X, d) be a metric space, r E X, e > 0, and E = {y E X : d(x, y) < e}. Show that E is closed.

A: Let (X,d) be a metric space, with x∈X , ε&gt;0, and E={y∈X:d(x,y)≤ε} To show: E is closed. That is…

Q: If (X, τ) is a topological space and if A,B⊂X Show that ∂(A∩B)⊂∂A∪∂B.

A: Boundary of a set in topological space: Let A be a subset of a topological space X, τ. The boundary…

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

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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: Let x1 and x2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

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Q: If A is a compact subset of a metric space (X, d) and B is a closed subset of A, prove that B is…

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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: A° U B° c (A U B)

A: