Define a collection T of subsets of Z+ as follows: W ∈ T if and only if n ∈ W implies that all positive divisors of n are also elements of W. Verify that T is a topology on Z+. In this topology find Cl({1}) and Cl({2}).

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A: a. The Hasse diagram is shown below.

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A: In topology and related branches of mathematics, a connected space is a topological space that…

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Q: A filter F on S is an ultrafilter if and only if it is maximal.

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Q: Show

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

Q: Let X = {a, b, c, d, e}. Create a σ-algebra X from set X. Explain why it is a σ-algebra.

A: Given : The given set is  X = {a, b, c, d, e} To find : We have to create a σ-algebra from the set…

Q: (3) X is not the union of two disjoint non-empty separated sets. (4) X is not the union of two…

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Q: If S is a subset of T , then u* (T) su* (S).

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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

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Q: If M and N are two norms on space X then prove that M and N are equivalent if and only if a subset…

A: Any two norms M and N on a norm space is called equivalent if  M≤N≤M. in the given question we have…

Q: Let M, and M2 be two R-modules and N, and N2 be two submodules of M, and M2 respectively. Also, let…

A: Here we need just one fact that if f:X→Y is a function defined from X to Y and if y∈Y, then f-1(y)∈X…

Q: (a) Suppose S is a subset of positive integers, ZZ. If |S| ≥ 3, prove there exist distinct z and y…

A: (a) Given that S is subset of  such that Observed that any positive integer is either even or odd…

Q: Let X be an infinite set endowed with the finite complement topology T, then (X,T) is compact. * O…

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Q: The intersection of any countable family of

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Q: (a) When is a subset of a topological space said to be closed? (b) Fotmulate the definition of…

A: If X,𝞽 is a topological space, then a closed set in X is a subset of X whose complement is a member…

Q: B) Show that every countable discrete family {F} of finite subsets of T3, then there exists a family…

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Q: Let 2 be the universal set and let G= p(2) the power set of 2 Define the follov operations X+Y =…

A: Given: Let's take Σ as the universal set and let G=ρ(Σ), the power set of Σ. X+Y=X∪Y;XY=X∩Y;Xc=X.

Q: (c) Prove that only Ø and R are open subsets of R

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Q: Assume that A is a σ-algbera of subsets of ℝ. Prove: If A contains all intervals of the form (a,∞),…

A: A σ-algebra on a set X is a collection of subsets of X that is closed under complement and countable…

Q: A is an ordered pair (G,*), where G is a nonempty set and is a binary operation on G.

A: We can solve this as follows:

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A: Given that Z is a set of binary space which have eventually zeros. This means that it contains zero.…

Q: Let (X. T) be a T, topological space and A a nonempty subset of X, if x is a cluster point of A…

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Q: If A and C are nonempty classes, prove that AC B and C C D if and only if AxC C BxD.

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Q: Consider the binary relation R = {(x,y),(x,z),(z,x),(z,y)} on the set (x,y,z}, which one of the…

A: We have to find which one of the options is true for relation R.

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: Let A and B be nonempty subsets of R such that A is bounded above and B is bounded below and let A –…

A: Let A and B are two non-empty sets of ℝ.Given A is bounded above and B are bounded below sets.Claim:…

Q: Give all σ-algebra generated by S = {0, 2, 4, 5, 7}.

A: Given that Give all σ-algebra generated by S = {0, 2, 4, 5, 7}.

Q: A space X is locally connected if and only if for every open set U of X, each comj

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