Let R (the set of real numbers) be equipped with the Euclidean topology, then A, a finite subset of R, is: O closed but not open in R O None of the choices neither open nor closed in R O open but not closed in R

Q: OConsider the subset A = {1,, ., , ..} of R, the set of all real numbers 1 1 with the euclidean…

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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: We will first show : (iii) If S is dense in X, then T is dense in X, where S and T are non-empty…

Q: Let A be an open subset of R. Is it true that (A)° = A?

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Q: • A is open in X. • A = Int A • A contains none of its boundary points. • Every point of A has a…

A: As per the company rule, we are supposed to solve one problem. Therefore kindly repost the remaining…

Q: Consider the following sets in the dictionarv order. Determine which of these sets are linear…

A: We know that Linear continuum is a linearly ordered set S of more than one element that is densely…

Q: 1.86 Prove or give a counterexample: Every open cover of a finite subset of R has a finite subcover.…

A: We need to determine whether given statement is True or False. Every Open cover of finite subset of…

Q: Which one of the following statements is true?

A: As in R with euclidean and R with finite closed topology then function f may not be continuous, thus…

Q: Cantor Intersection Theorem: Let {Q1 .Q2 .Qn be a countable collection of non-empty sets in the…

A: Given Q1,Q2, ..., Qn, .. be a countable collection of non-empty sets in the Euclidean space ℝn. such…

Q: A and B are bounded subsets of R, then prove that UB and AnB are bounded.

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Q: Va Abe B: a <b

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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: Every finite set in Euclidean topological space is closed

Q: Consider the set of real numbers, R. Let A = {x∈R:0≤x≤2} be a subset of R. Is the set A open,…

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Q: Let S be a set defined as S={x∈R∣−1≤x≤1}. Determine whether S is open, closed, both, or neither in…

A: In the standard topology on R, a set is considered:Open if for every point x in the set, there…

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: Consider the following topological spaces: Space X: A closed unit interval [0, 1] with the usual…

A: Two topological spaces X and Y are said to be homeomorphic if there exists a continuous bijection f:…

Q: Let R (the set of real numbers) be equipped with the finite complement topology, then A, an infinite…

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

Q: prove that If A, B, C are finite sets such that An B n c = Ø, then | A U BUC| = |A | + | B | + | CI.

A: Solution is following in explanation step.

Q: Which one of the following statements is true?

A: In the R with the Euclidean topology and R with the finite closed topology, f might not be…

Q: c) Let u be a usual topology on a set of real numbers R, and E= {2,3,6 Show that whether E is…

A: Given that R is endowed with the usual topology and that E={2,3,6}. We shall state a definition and…

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: Let A = {n€ Z| |n|<3} and the set B = {m/k[m€{2,6}, kE {1,2}}. Write the elements of set ANB in…

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Q: If U is homeomorphic to V and U is compact, then * O Vis compact V is complete V is connected O V is…

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

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

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: Ris not connected if T is * O None of the choices O the trivial (usual) topology the indiscrete…

A: R is not connected if T is

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

A: Answer

Q: Let A be a compact set in R. Show that the complement A is open and contains an interval of the form…

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Q: real numbers such that a < b. <x < b} is uncountable. <x < b} is countably infinite.

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Q: Suppose that R (the set of real numbers) is equipped by TO (the Euclidean topology), T1 (the…

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

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

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

A: Given that Let X be an infinite set and T be a topology on X.If every infinite subset of X is…