Consider two sets in ℝ² with the usual topology. Is the closure of a path-connected set also path-connected? Is the inverse true? (That is to say, if the closure of a set is path-connected, then is the set path-connected?)

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: If a metric space M is the union of path-connected sets Sa, all of which have the nonempty…

A: Given M is a metric space which is union of path connected sets Sα. All Sα have the non empty path…

Q: Give an example of a set in R ^ 2 with the usual topology that is not connected but has a connected…

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

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

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Q: 2- Prove that H, is convex set.

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Q: -prove that for any bounded Set E, There is abounded set A That is a countable intersection fopen…

A: The solution is:

Q: If A C BỊ U B2 where B1 and B2 are disjoint open sets and A is compact, show that An B1 is compact.…

A: To Show- If A ⊆ B1 ∪ B2 where B1 and B2 are disjoint open sets and A is compact, show that A ∩ B1 is…

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

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Q: Q.A. Is the nonnepatbe orthaut t. a cohvek set?

A: Answer is negative

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

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Q: - If C is Convex set ond a GR, then XC is a Convex set

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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

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Q: Decide whether the following propositions are true or false.If the claim is valid, supply a short…

A: A compact set is always closed and bounded.

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Q: Is there a set A that satisfies A = {A}? If yes, exhibit one such. If not, Why not ?

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

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Q: In R with the cofinite topology, every preopen set is open

A: As we know that , In ℝ with discrete topology of every semi open set is open.

Q: Let {S,}EN is a disjoint seqeuence of measurable sets , then: US, µ(S,)

A: Given a statement

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: A set L cR is convex if:

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Q: Đ A is a closed set.

A: We prove by using the result that intersection of two closed set is closed.

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: let X={a,b,c} Construct two topologies for the set that have at least 4 open and that are…

A: Given that X=a,b,c. To construct two topologies that have at least four open sets. We know that in…

Q: Prove that the open ball B(p, 8) = {x : ||x – p|| < 8} is a convex set. [Hint: Use the Triangle…

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Q: 4. Prove that the set Z is a closed set in R.

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Q: is the set [5,100] compact?

A: Given that:A set is S=5,100.We have to check whether a set 5,100 is compact or not?

Q: Determine if the set S is span set of %3D R2x2. Solution:

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Q: Show that (B - A)U(C- A) = (BUC)- A 1) Using a Venn diagram 2) Using the definition of equal sets

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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: The cardinality of a finite set A, denoted by Al, can provide the of A as equal to 21A

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Q: ndary of a path-connected set in R^2 with the usual

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Q: Give example for following 1/ set is not compact

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Q: Prove that a set A is connected iff A is and interval.

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Q: A collection of open sets in R with the standard topology whose intersection is not open.

A:

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

A: