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Q: Show that every second countable (A2-space) space is separable.

A:

Q: Show in the discrete metric space: 1. All sets are open (especially single points);

A: Let X be a non-empty set and d be discrete metric on X

Q: Ametric space (x,d) is complete if and only ifx has the cantor intersection property

A:

Q: Let (X,T) be a toplogical space, A, BC X. Show that АС В Ас в, АсВ. 介

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

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Q: what are domine, Vange and null space o F Tisra-s Ty Rom R?into Re 1

A: The given functions are                                 T1:ℝ2→ℝ2 such that T1ξ1,ξ2=ξ1,0.T2:ℝ2→ℝ2…

Q: Show that the space (P is complete.

A: The given problem is to show that the space lp is complete. we can use properties of lp spaces to…

Q: ExeraiSE 4Let the Open set dEnse 5- YOVE metrc Space Cs,d) Hen in

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Q: Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

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Q: Prove it. Establish the mapping 7 and IN.

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Q: Show that an infinite intersection of closed sets F, k = 1, 2, 3, . .., in a metric space (S, d) is…

A: first we will proof that an infinite union of Open sets in metric space ( S, d) is open using the…

Q: In a metric space (M, d), show that both Ø and M are open sets in M. In a metric space (M, d), show…

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Q: Let (X.d) be mnrtric space and X is closed then X is O complete Ono decsion Onot complete

A: A metric space X is said to be complete if every cauchy sequence in X is convergent. That is, every…

Q: PROVE THE FOLOWING: PROVE thtAT IF AND ONLY %3D IF is REAL

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Q: Determine whether is a linearly independent set of vec- 3 11 tors in R3.

A: A set of vectors v1,v2,v3,…,vk in a vector space is said to be linearly independent if…

Q: Let X be a topological space. Prove that (a) Ø and X are closed. (b) Intersections of any number of…

A: Union of any open sets is open and intersection of two open sets is open.

Q: Be M, N subsets of space W. Then span (MUN) = span (M) +span (N)- %3D

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Q: (a) Investigate whether d: RxRR delined as d(z, u) = (- u) is a metrie on the set ol all real…

A: We investigate given distance function is metric on R or not

Q: Let (x, eA) be a measurable space, £: X (-∞, w) and 9:* (-∞0, ∞o) are measurable. Prove that the…

A: Given that X,A is a measurable space. The objective is to prove that the sets  {x∈X:f(x)<g(x)},…

Q: Call a set A CR clopen if it is both open and closed. Prove that the only clopen sets are Ø and R.

A: A⊆R clopen if it is both open and closed.  We have to prove that the only clopen sets are ϕ and R

Q: let ( X Td)be Toplog9 me trics spaces the n EVery singteon set is closed set

A: Given that X,Td is a Topology Metric spaces.

Q: Let (Ka)aeA be a collection of compact subsets of a metric space, X. Show naeAKa is closed in X.

A: A metric space X is compact if every open cover of X has a finite subcover.

Q: give example is not To-SPace about to folegical SPace

A: We have to find a topological space which is not t0.

Q: ket (X, @) be a topokgical space, Ka compact subset of X and (FalneN a family in PCX) with Fr+ Ø POR…

A:

Q: Let R be the set of all real numbers with the euclidean topology. Show that R has an uncountable…

A: Let R be the set of all real numbers with the euclidean topology. Show that Rhas an uncountable…

Q: Let X And Y be two distrecte spaces. Then X is homeomorphic to Y if and only if X and Y are both…

A:

Q: LBe M, N subsets of space W. Then span (MUN) = span (M) + span (N)“ %3D

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Q: Use the definition of compact set to prove that the set [0,1] is compact

A: We know that a set E is compact if every open cover of E has a finite subcover. We suppose, that O…

Q: Let X be an infinite set, then any topology on X is also infinite. Select one: O True O False

A: Note: " Since you have asked multiple question. As per our guidelines we are supposed to solve only…

Q: Consider the subsets [0, 1] and [3, 4] of R, the set of all real numbers with the euclidean…

A: Consider the subsets [0, 1] and [3, 4] of R, the set of all real numbers with theeuclidean topology.…

Q: 5. Describe the closed sets in the following topological spaces: a) (X, E,) wherepe X. b) (X, P,)…

A: (a) Given,           (X, Ep), where p∈X

Q: Prove that intersect on of two sees Sets Sets Set? lonve ond hence ony number of lonved. what ebout…

A: To show:  Intersection of two convex sets is convex   Let A, B be two convex sets  We need to show…

Q: Let ACR be a nonempty bounded set. Prove that sup A is unique.

A:

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: Iwant to define these Poin (Defined) 1-) sobolve spaces 2-) Reflexive real Banach SPace

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Q: Remark: If (V,) is an inner product space. Then for all uiv,we Vand a,p EK い = &+p リ = す

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Q: Show that the interval [0,1] is a closed set in R

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Q: Let M# ¢ is subset of a Hil bert TT H. Show that M is the Smallest Space closed Space. of H which…

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Q: Show that the set of all real number constitutes incomplete metric space an we choose doxoy) = Itan'…

A:

Q: Prove disprove that (R.Tco-finite) is Te-space or

A: To Prove - Prove or Disprove that ℝ, τco-finite  is T2 - space.       Definition used - T2 - space…

Q: tet is cosinte toploy on set R show that (R, t) is O NOVmal space- Regualra space

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Q: ii). what { disie [4NI} to fobrm vedtor space ? set vectas a bauis for en N-dimeneional alte the…

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Q: Let the non-empty set S = {x - yv3:x,y ER}, show that S is closed'under multiplication then find the…

A: We have to show that the given set S=x-y3:x,y∈R is closed under multiplication and also find the…

Q: a nonempty compact set of IR. sup Ke K. be

A:

Q: let E be a set that baunded. Prove that Inf E = Sup -E) where R, XE E y

A: To prove for a buonded set E inf(E) = -sup(-E) , where -E = {-x, x belongs to E} , we first prove…

Q: Let X be infinite set and x0 ∈ X. Consider the topology τ = {G ⊆ X : x0 ∈ G or G = ∅} on X. Is (X, τ…

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Q: )Š is the smallest closed set in M such that S CS.

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Q: Remark: If (V, = « +p vi " = a + p

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Q: Prove that S UT = SN T for all sets S and T using the rules for complements of sets.

A: rules of complements x∈A⇔x∉A¯