6. Show that the span of this set is all of R?. {}}

Q: If  X is a countable set show that every topology on X is second countable.

A: To Show: If  X is a countable set show that every topology on X is second countable.

Q: Show that S CR is a closed set if and only if S contains all of its limit points.

A:

Q: Show that R has the same cardinality as the interval (–2, 2) by building a bijection between the two…

A: This is a problem of Real Analysis and Topology.

Q: 5. Let E be a set that bounded and c > 0. Prove that sup(cE) = -c inf(-E) %3D

A: This is a problem of Real topology, metric topology.

Q: show that the Cantor set is a closed subset of R and has measure zero.

A: We know the definition of Cantor set, The Cantor set is defined as the intersection C = ∩n=1∞Cn ,…

Q: Choose any that is a subset of the set S={10, Ø,{{ Ø}}} 0{0} {(Ø), 10) 0 {{0}}

A: Solution

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A: Let's examine the  problem for path-connectedness example of a set in R2 with the usual topology…

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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: 1) Define a subset.

A: To define a subset.

Q: 3.1 Show that a finite intersection of closed sets is closed.

A:

Q: C) = (*,') + "- ), Give an interpretation involving subsets. Problem 4) Prove that

A: See the hand written

Q: S is a nonempty set of real numbers that is bounded.

A:

Q: The continuous image of a connected set is connected. That is, if X is connected and f: X→ Y is…

A:

Q: 6. Every finite set in R is compact. Why?

A: The given question is related to real analysis. We have to show that every finite set in ℝ is…

Q: Define Closed sets ?

A: Closed sets are defined as a set which consists all of their limits points. It is also defined as…

Q: Choose any that is a subset of the set (Ø} {{Ø),0} {{1}}

A:

Q: Prove that if S is a bounded set then S is bounded.

A:

Q: Prove that empty set is subset of all sets: φ ⊆ A

A: Given: An empty set and set A is given. To Find: To prove that empty set is subset of all sets: ϕ⊆A

Q: Q.Prove If ANB is pathwise connected set, then A andB are or disprove the following: pathwise…

A:

Q: 6 Prove that a closed set in the metric space (S, d) cither is nowhere dense in S or clse contains…

A:

Q: & and X are open sets.

A:

Q: A set which has ho element is said to be

A: A set with no elements = { }

Q: 4. Let A be an open set in å nörmed space subset B of X .

A: Que  Ans :  Given ( X, ||-||) be a norm linear space,  A be any open subset of X    ( def- set A is…

Q: Every connected subset of R is complete is ture or false?

A: Every connected subset of R is complete is true or false?

Q: Is the Cantor set closed? Is it compact? Explain!

A:

Q: Give an example of a closed set in R ^ 3 with the usual topology that is not compact

A: Given the usual topology on ℝ3. i.e. τusual=U⊆ℝ3: ∀x∈U, ∃∈>0 such that B∈x⊆U

Q: The continuous image of a connected set is connected. That is, if X is connected and f : X–→ Y is…

A: We prove this via contradiction. We assume that f(X) is not connected then X is not connected. And…

Q: shows that there is no topology in X based on the family B = {{a, b}. {a, b, d}, {b, d, c}}

A:

Q: If G(sub alpha) is a collection of closed sets, then the intersections of the G(sub alphas) is a…

A:

Q: Let {S,},EN is a disjoint seqeuence of measurable sets , then: HUS,

A:

Q: Show that the derived set E′ of any set E is closed.

A: Here is the proof .

Q: Which of the following sets are closed under addition? Choose ALL that apply.

A: A set A is closed under addition, if for every x and y belong to A then x+y also belong to A.

Q: Show that any subset of bounded set is bounded

A:

Q: The closare of a set is smallest closed set containing s صواب ihi

A: Let S be a subset of X: S⊆X. Let M be the closure of S. Then M is the smallest closed set which…

Q: A set L cR is convex if:

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Q: 7. Show that the Obenchain class LOb(B) is a convex set.

A: Given: To show the Obenchain class LOb (β) is a convex set is given as,

Q: show that empty set, X is an element of S intersection T in topology

A: Here We have to show that empty set must be in S∩T where S and T are topology on set X.

Q: Construct a family of sets that demonstrates R is not compact

A:

Q: 7 The set M={xEN:x<÷} with the usual metric is connected.

A:

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

A:

Q: 7. Suppose that a set X is equipped with the finite closed topology. Which one of the followings is…

A: As per the policy, we are solving first question. Please repost the question and specify which…

Q: Show that X is connected if and only if the only clopen subsets are Ø and X.

A:

Q: 5. State whether each of the following statements is true or false, in either case substan- tiate/…

A:

Q: Give an example of a set in R ^ 2 with the usual topology that does not have any subset (not…

A: Given that R2 with the usual topology. The aim is to find an example of the set  R2 with the usual…

Q: Let S be a set of n elements. Find the number of distinct subsets of S.

A: Given: Let S be a set of n elements.

Q: 3.1 Show that an infinite intersection of closed sets F, k 1,2, 3,..., in a metric snace (S, d) is a…

A:

Q: Show that Z has the same cardinality as the set E of even integers.

A: In order to prove that ℤ and set of even integers E have the same cardinality. (i.e) we need to find…