Let R (the set of real numbers) be equipped with the discrete topology, then the semi open interval ]1,2] is O open O Closed Clopen O Neither open nor closed

Q: Let A and B be subsets of topological spaces X and Y, respectively. Prove that CI(A × B) Cl(A) ×…

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Q: 5) Let R be the set of all real numbers with the euclidean topology. Define the set A = {(x, y) E R…

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Q: Let X be any set with the discrete topology. Show that X is normal.

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Q: Let T be the finite closed topology on a finite set X, then * O Tis the Euclidean topology O Tis the…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Let A be any set with the discrete topology. Show that A is normal.

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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: * Let R be with the discrete topology. If A = {1, 3,5, 7, .}, then Aº A O N O RO

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Q: Let R (the set of real numbers) be equipped with the Euclidean topology, then A, a finite subset of…

A: Given that R is equipped with the Euclidean topology and A is a finite  subset of R.

Q: Let X be a topological space. Prove that (a) Ø and X are closed. (b) Intersections of any number of…

A: Union of any open sets is open and intersection of two open sets is open.

Q: 2. Let f be defined on an open interval (a, b) and assume that for some e e (a, b) we have f'(c) > 0…

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Q: The collection of all closed intervals forms.( a subbase for a topology on R. This :topology is…

A: The collection of all closed intervalsfrom a subbase fo a toplolgy on RThis Topology is :-consider…

Q: Let X = R and s be a collection given by ( = {0}U{0cX:0° <x}. (a) Show that S forms a topological…

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

Q: Let R (the set of real numbers) be equipped with the Euclidean topology, then A, a finite subset of…

A: First option is correct

Q: (d) Consider the set [1, √5] C [-1, √10]. Explain f is not continuous by (i) connectedness, (ii)…

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Q: 1 Let B be the set of all bounded sequences of real numbers and define the function d: B x B → R by…

A: we have to solve given problem:

Q: Prove that the set {x e R : 10/x - x > 0} is bounded. Prove that the set {x E R : x² – 25x > 0} is…

A: To prove that the set x∈R:10x−x&gt;0 is bounded. To prove that the set x∈R:x2−25x&gt;0 is unbounded.

Q: Which one of the following statements is true? * R with the Euclidean topology and R with the…

A: Two homeomorphic spaces share the same Topological property If one of them is compact, then the…

Q: The set X is an open set in the ambient metric space . O True O False

A: Solution

Q: Consider the particular point topology, PPX, on a set X. Determine Int(A) and Cl(A) for sets A…

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Q: Let R be the set of all real numbers with the euclidean topology. Show that R has an uncountable…

A: Let R be the set of all real numbers with the euclidean topology. Show that Rhas an uncountable…

Q: Let X be an infinite set with the countable closed topology T-{S subset of X; X-S is countable}.…

A: Given that X is an infinite set with the countable closed topology T-S subset of X; X-S is…

Q: Ler be a topology on X. Assume that is Husdouff and let xe X. 1. Show that {x} (and hence every…

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Q: Let R (the set of real numbers) be equipped with the Euclidean topology and let S = {0, 1, 1/2, 1/3,…

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Q: Let T be a topology on X. Assume that T is Hausdorff and let x € X. (1) Show that {x} (and hence…

A: let T be a topology on x. T is a Hausdorff space

Q: an open ball in the set X, with its radius & satisfying -d(x, a), Vx € X at the open ball Be (x)…

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Q: Let R (the set of real numbers) be equipped with the Euclidean topology, then A, a finite subset of…

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Q: A. Show that T is a topology in X and Show that B,(x) is open set with respect to t for all r > 0

A: This is a problem of Topology.

Q: Let X be an infinite set with the finite closed topology T={S subset of X; X-S is finite}. Then

A: a) If not, then there are non-empty disjoint open sets  such that  . Since they are disjoint, so  .…

Q: Consider the number line R with the topology t = {Ø, R, (-0, a); a E R}. Is the set A = (-3, 0]…

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Q: Let R be set of real numbers equipped with the standard topology. Clarify with full explanation…

A: Heine- Borel:- A subset of ℝ is compact if and only if it is closed and bounded. Sequential…

Q: 1.Any open interval in R is an Open set in R 2.The {0} is open in R with discrete metric

A: We will use the fact of open ball

Q: 3) Consider the topology 7 = {G CR: 0 ¢ G or G = R} on R. Is open T 3= interval (-1,1) compact in…

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Q: (d) Consider the set [1, √5] C [−1, √10]. Explain f is not continuous by connectedness, compactness,…

A: Given: The function, f:-1,10→ℝ defined by, fx=x2,x&gt;2x,x≤2 The set 1,5⊂-1,10. To explain: Why the…

Q: Let X be an infinite set and T be a topology on X. If every infinite subset of X is in T, then T is…

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Q: Let X be an infinite set and I be a topology on X. If every infinite subset of X is in T, then Tis *…

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Q: Let F and G be nonempty bounded sets. Prove that inf (F UG) = min{inf F, inf G} and sup (F UG) =…

A: Since F and G are non-empty bounded subsets of R, sup F, sup G, inf F and inf G are all exist in R.…

Q: 3) let X be discrete topology and Ac X, find A ?

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Q: Let T be the finite closed topology and be also the discrete topology on a non empty set X. Then…

A: X is finite

Q: Prove that if A is not connected, then A is not an interval.

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Q: 2) Let R be the set of all real numbers with the euclidean topology. Show that R has an uncountable…

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Q: Consider the topology 7 = {GCR:0 ¢G or G = R} on R. Is open interval (-1, 1) compact in (R, T)?…

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Q: Let R (the set of real numbers) be equipped with the Euclidean topology, then A, a finite subset of…

A: Solution-

Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is F closed?

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Q: |A:k||2||Br:||2) < ||A||F||B||F k=1

A: answer is in next step

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: 5) Let R be the set of all real numbers with the euclidean topology. Define the set A = {(r, y) e R…

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Q: Which one of the following statements is true?* R with the Euclidean topology and R with the…

A: Which one of the following statement is true

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

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Q: - A non-empty set E in R is connected if and only if it is an interval

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Q: Let R (the set of real numbers) be equipped with the Euclidean topology and let S - (0, 1, 1/2, 1/3,…

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