Prove a neighborhood is an open set

Q: Prove that a subset of a set of measure zero has measure zero.

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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: 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: In the discrete topology on the set X, every set is open and it also follows that every set is…

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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: 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: Prove that the smallest member of a set, if exits, is the greatest lower bound of the set.

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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: Suppose a set M is closed and bounded. Prove that M has a maximal and minimal element.

A: Suppose a set M is closed and bounded. Then we have to show that M has a maximal and minimal…

Q: Prove or disprove. A closed and bounded subset of e' is compact.

A: PROVE OR DISPROVE :-  A closed and bounded subset of l1 is compact.

Q: Provide a counter example that infinity intersection of open set is open

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Q: that the boundary of a closed set is nowhere dense in

A: Matric space

Q: (b) Show that the continuous image of compact set is compact but the convers need not to be true.…

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Q: Prove that any closed interval [s, t] can be written as the intersection of a collectable collection…

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Q: Show

A: We prove this statement by using definition of Borel Set.

Q: (c) Prove that a metric space is connected iff it contains exactly two sets that are both open and…

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

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Q: Use the Heine-Borel theorem to give an example of a set not compact in R with the usual topology…

A: An example of a set not compact in R with the usual topology and a non-compact set R2 with the usual…

Q: Give example open set is compact

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Q: prove that Cantor set is closed and compact.

A: Use the property that arbitrary intersection of closed set is closed and the Heine Borel Theorem.

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: 4. Prove that the intersection of an arbitrary collection of compact sets in R is compact.

A: (Heine-Borel Theorem). A subset A of R is compact if and only if A is bounded and closed (with…

Q: & and X are open sets.

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Q: Prove: The Cartesian product of countable sets is countable.

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Q: Show that any convex open set is simply connected.

A: The objective is to prove that any convex open set is simply connected. Simply connected is nothing…

Q: In any metric space, the only open and closed set at any time is the empty set and X only True False…

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Q: Find a topology in which the point x = 5 is an open set.

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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: Prove that every Union of Open sets is also Open.

A: To prove: Every Union of Open sets is also Open.

Q: Prove that the Cantor set has measure zero.

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Q: Suppose A is a set and R is a reflexive and transitive relation on A. Show that the

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Q: Prove that the greatest member of a set, if exits, is the least upper bound of the set.

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Q: prove that empty set is a subset of every set

A: We have to prove that empty set is a subset of every set = ?

Q: Show that any compact set in R is closed and bounded.

A: Proof

Q: Prove that a set is closed if and only if its complement is open

A: prove of the given problem is given below...

Q: Prove that a nonempty set is bounded if and only if it is contained in a bounded interval

A: A non empty set S is called bounded if it has both upper and lower bounds, In a certain sense a non…

Q: Prove that every nonempty subset of a well-ordered set is well-ordered.

A: Given problem is :

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

A: Here is the proof .

Q: Show that any subset of bounded set is bounded

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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:

A: In this question, we have to find the correct option for the convex set L⊂Rn  here we use the direct…

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

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Q: Prove that in any metric space, an ɛ- eighborhood of a point is an open set.

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Q: Every proper simply connected oper equivalent to an open subset of the

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Q: open ball is the open set

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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: B. Prove that the union of a finite number of compact subsets F1, F2, ..., F, of RP is a compact…

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Q: Prove that the set of all finite subsets of ω is countable. (Set theory)

A: The objective is to prove that the set of all finite subsets of ω is countable.