Let x be any set then the indiscrete topolagy ■ is coarser that any topology I defined on x- 5 Let x be any set then the discrete topology is finer that any topology I defined on x. التاريخ الموضوع : let R be used of real numbers then the usual topology on R finer that cofinite topology on R Let IN be aset of natural numbers then the Cofinite topology on (I) finer than final segment. on IN. (c

Q: 11. If a set has empty interior, then so does its closure. 12. The closure of a set coincides with…

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Q: 11. If a set has empty interior, then so does its closure. 12. The closure of a set coincides with…

A: We need to show whether the given statement "If a set has empty interior, then so does it's closure"…

Q: Select all that apply: The set II-1(-j,j) as a subset of X = R" is open in - The Discrete Topology…

A: In the given question we have to tell about the set s s.t why the set  is open in Discrete topology…

Q: 19. Let X be any nonempty set, and let B = {{x}: x € X}. Show that B is a base for a topology on X.…

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Q: yes, hi. I've seen another source, use set variables to 0 and then solve individually, answers are…

A: setting z=0 and solving for x and y. similarly setting y=0 and solving for x and z.

Q: Theorem 1.14. The set of all finite subsets of a countable set is countable.

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Q: For each of the following collections of sets, define a set An for each n N such that the indexed…

A: (a) From given collection of sets,we can say ,In other words, given collection is nothing but now…

Q: Segment congruence theorem. Segment congruence is an equivalence relation. In other words, if PQ, RS…

A: If PQ≅RS then by definition, d(P,Q)=d(R,S)...(equation 1) and if RS≅UV then by definition,…

Q: xercise 2. Show that a set with the discrete topology is compact if and only if it is finite. if and…

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Q: 3) #36) Hint: {[a,b): a <b} then the Borel o-algebra this complete the pr

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Q: Let X be any nonempty set, and let B = base for a topology on X. Describe the open sets of this…

A: Given: Here is the given data, X be any nonempty set, and B= x:x∈X  To show: B is a base topology on…

Q: topological space a

A: Presumably, the closure of Q in R under the usual topology. It suffices to show that for every real…

Q: 1. Consider the sets D = { // : n € J}, E = DU{0}. (a) Really quick: complete the definitions - (a)…

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Q: For which spaces ++ is it true that covery single ton set is closed

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Q: Let A = {Ba: a € A} be a family of sets and let G be a nonempty set. Prove Gn an (UB.) - U(GOB₂) =…

A: Given: A=Bα : α∈∆ be a family of sets and let G be a nonempty set. To prove: G∩∪α∈∆Bα=∪α∈∆G∩Bα

Q: Exercise 2.7. Give an example of a topological space and a collection of open sets in that…

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Q: when can two spanning sets be equal?

A: See solution below

Q: (c) Find an open cover of D that does not have a finite sub-cover. Prove it. (d) Suppose UE is an…

A: A subset is compact if every open cover has a finite subcover.