let C be a compact subset Tn a Haus dorft topolopfcal space (K,R) and bex a If bac then prove that there exist disgoint point. open sets U and U such that bEU and C

Q: A topological space is Hausdorff iff limits of all nets in it are unique

A: This is a problem of Topology.

Q: Define the closure of a sot I in a motric spack. How is the boundry it a sut I related to it clogwai

A:

Q: Prove that in a metric space (S,d) every closed ball Sr[Xo] is a closed set

A:

Q: For any topological space (X, T) and A C X, the set A is closed. That is, for any set A in a…

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

Q: Every Hausdorff space is hereditarily Hausdorff.

A: We have to prove that every Hausdorff space is hereditarily Hausdorff. Consider a Hausdorff…

Q: Let (X,T) be a separable topological space, let A be an open subset of X. Then (A, T,) is a…

A:

Q: UNA annetect in X show that Tivi is a connected set in a space Y. ) Let T be a cofinite topology…

A: If  X is a non-empty set, then the collection of subsets of X whose complements are finite along…

Q: A normed space X overR is separable if it has a conutable dense subset 0 صواب ihi

A: The given statement that " A normed space X over R is seperable if it has a countable dense subset "…

Q: Cantor Intersection Theorem: Let {Q1 .Q2 .Qn be a countable collection of non-empty sets in the…

A: Given Q1,Q2, ..., Qn, .. be a countable collection of non-empty sets in the Euclidean space ℝn. such…

Q: 1) Every convex set of a vector space X is affine set.

A:

Q: Let the definition of disconnected be the definition in the picture attached. Show that this…

A:

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: Let

A:

Q: Let X be a non empty set and p be a fixed point in X. Define T={X, ACX such that p€A}. Then any…

A:

Q: a. Is there any relation between reflexive normed space and a Banach Space? (If yes then prove) b.…

A: Since you have posted multiple questions but according to guidelines, we will solve first question…

Q: (BOunded st) show that a subset M in a nor med space Xis bounded if and only If there is a positive…

A:

Q: Read the graph pooot of the closed theorem carefully ounel explain wher nd how we have used facts in…

A:

Q: rove that spaces Ip for os p<l If for pe(0, 0) {0} and complet. are

A:

Q: Let (x,c) be a topologıcal space. Property C=P.C: A subset A of X has P.C if it's subset of the…

A:

Q: Prove that any reflexive normed space is complete

A: A normed space X will be said to be complete if each Cauchy sequence in X is convergent in X.

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

A:

Q: on 'N" en Prove Delween Aborms is lenre equala bub

A:

Q: Letxandy and z be normed space s and s: xy andTIYz bounded oper atorsprove

A:

Q: let X be a firite dimensional notm Space Then Prove that is compat.

A: As X is a finite dimensional norm space, this means that, dimX is finite.

Q: ## Commpact subse regular Space X and if of is subset of A such that .cey th there is a closed a…

A:

Q: Suppose A is a closed, non-empty subset of a metric space (X, d) and p a point of X which is not in…

A: Given:  A is a closed and non-empty set of a metric space (X,d) and p∈X , p∉A. To Prove: inf{d(p,a)…

Q: Prove that closed balls are closed sets in the standard topology on R?.

A: Topology Question

Q: Use the definition of compactness to prove that a closed subset Y of a compact set XCR is compact.

A: We say a set K⊂R is compact if every open cover of K has a finite sub cover.

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: 6. Prove that if x1, x2,..., Xn are distinct points in a Hausdorff space X. Prove that there are…

A: Let X be a Hausdorff space. Then in X any two distinct points are separated by disjoint open sets.…

Q: ANB +P

A:

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.…

A:

Q: 3: Prove that any reflexive normed space is complete.

A: