If B1,..., Bn is a finite open cover of a compact set A, can the union B1 U ·. U Bn equal A exactly? ...

Q: Let A be a set. Suppose that f: w→ A is onto A. Prove that A is countable.

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Q: Let S be a nonempty subset in R". Prove or disprove by a counterexample that Span S = (S+)+

A: We have to prove or disprove by counterexample that Span S=S⊥⊥

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

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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: the collection of all close subsets of R is called the topology of R

A: To check: Whether the statement is true or false: Given: The statement is "The collection of all…

Q: For any collection a of subsets of X, there i ntaining a.

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Q: Prove that the complement of a dense open subset of R is nowhere dense in R

A: Prove that the complement of a dense open subset of R is nowhere dense in R

Q: Let A be a subset of a topological space X. Show that bd(A) = Ø iff A is open and closed in X.

A: We need to prove the statement.

Q: Let A1,·.., Am be compact subsets of R. Prove that Aı × · … × Am is a compact subset of Rm.

A: Given, A1,A2, ... , Am are compact subsets of R. We prove A1x A2 x ... x Am is a compact subsets of…

Q: m) Show that if T;,T; and Ty are topologies on a set X.then r, nnt, is a topology on X.

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

Q: 4.5 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

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Q: Show that the set K={a, b} with actions + and - defined tables   is a body whose zero element is a…

A: We are the set K={a, b} with actions + and ∙ defined in the following tables. Notice + operator…

Q: the collection of all close subsets of R is called the topology ofR ylgn İhi

A: the collection of all close subsets of R is called the topology of R

Q: If B1,..., Bn is a finite open cover of a compact set A, can the union B1 U... U Bn equal A exactly?…

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Q: 3.6 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

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Q: 1. Determine if the given collection of subset of X={a, b, c} is a topology on X. a. T1 = {Ø, X,…

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Q: Prove that for any infinite index set J, the uniform topology on R° is strictly finer than the…

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Q: 4. Use the definition of compactness to prove that a closed subset Y of a compact set X CR is…

A: We will answer the first question since we answer only one question at a time. Please resubmit the…

Q: Let (X, T) be a topological space, AcX, then A'next (A) = 10 AO

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Q: Let {An : n e N} be the indexed collection of sets defined by (2. 2-) 1 An Prove that: = Ø

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Q: The set S=[1,3) is not compact. Which open cover of S contains a finite subcover?

A: This is the question of Topology.

Q: Suppose f, g, and h are all mappings of a set A into itself. a. Prove that if g is onto andf 0 g = h…

A: Given that, f,g and h are all mappings of a set A into itself. If g is onto and f∘g=h∘g  To prove…

Q: Prove that every finite set S of R" has no accumulation point.

A: Now taking set S Let ? = {?1, ?2, . . . , ?n }.

Q: Let (X.J) be a topolgical space and {C1, C2 Cn} be a finite collection of compact subsets of X.…

A: Given Let (X,I) be a topological space and C1,C2,......Cn be a finite collection of compact subset…

Q: Let A be a subset of some universe U. Prove that AN A° = Ø.

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Q: . Show that every map f : A → B of sets decomposes as hog where g : A → C is surjective and h : C →…

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

A: Use the following definitions, to prove the result. If a set contains all its interior points, then…

Q: Theorem 6.17. Every compact subset C of R contains a maximum in the set C, i.e., there is an m E C…

A: To prove that, every compact subset C of ℝ contains a maximum in the set C, that is, there is an m∈C…

Q: Let {An : n E N} be the indexed collection of sets defined by 1 2, 2+ 2n An Prove that: nMeN An = Ø

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Q: Let {Aα:α∈I}{Aα:α∈I} be a nonempty indexed collection of sets.  Prove or disprove…

A: Let Aα:α∈I be a non empty indexed collection of sets.  If A is a family of sets, the intersection…

Q: Amongst the various possible products of four topological spaces A, B, C and D, show that (A × (B ×…

A: We shall solve this in next step

Q: show by an example that a nonempty proper subset of S in the metric space (S, d) can be both open…

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

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Q: (b) Prove that the complement of a dense open subset of RR is nowhere dense in R.R..

A: (b)  Prove that the complement of a dense open subset of R(R) is nowhere dense in R(R).

Q: Let X = {a, b, c}. Is the collection {X, Ø, {a}, {a, b}, {c}} a topology on X?

A: Here, the given set is X=a,b,c and collection is X,∅,a,a,b,c.

Q: 6. Prove that if M is a o-algebra on a set X and u is a comple: measure on M, then μ(X)<∞.

A: Let M be a σ-algebra on a set X. A complex measure on M is a complex-valued function μ:M→ℂ…

Q: (b) Prove that the complement of a dense open subset of R is nowhere dense in R..

A: Prove that the complement of a dense open subset of R is nowhere dense in R.

Q: A topological space (X, t) is a T1-space if and only il for each x e X, the singleton set {x} is…

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Q: For nonempty sets A, B, and C, let f: A -> B and g: B-> C. If g is onto, then gof is onto.

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Q: Theorem 2.20. For any set A in a topological space X, the closure of A equals the inter- section of…

A: Let A  be a set in the topological space X. To prove that the closure of A  is the intersection of…

Q: Consider the subset G {r• /2: x € Z} of R. Show that G is closed un- der addition.

A: The given subset is: G= x·2 : x∈ℤ

Q: An indexed family of sets (A,hcs is said to be a cover of a set X if and only if X CU,crA Use this…

A: Let A be a non-empty set and Cii∈I and Djj∈J are two different covers of A. Since Cii∈I is cover of…

Q: the collection of all close subsets of R is called the topology of R ylgn İhi

A: False

Q: Let {Fa}aeA be a collection of closed subsets of R. (a) Prove, from the definition of closed, that…

A: Given that Fαα∈A is the collection of closed subsets of ℝ. The objective is: To prove that ∩α∈AFαis…

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: Show that every closed subset F of R' is the intersection of a countable col- lection of open sets.

A: detailed explanation mentioned below.

Q: 3. Let A and B be arbitrary sets. Use double inclusion to prove that (A U B) N (A U B') = А.

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Q: If K is a compact subset of RP and z is a point of RP, then the set K, = {r+y :y € K} is also…

A: Given that K is a compact subset of ℝp and x is a point of ℝp. The objective is to show that…

Q: If (X,, T,), (X, T2) are two topological spaces, and A,, A, are open sets in X,and X, respectively,…

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