Let X be a topological space; let A be a subset of X. Suppose hat for each x ∈ A there is an open set U containing U ⊂ A. Show that A is open in X.

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

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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: A set X and a topology on X which is not Hausdorff.

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Q: Let A and B be two subsets of E. Prove using the set identities: А - В — В — А.

A: You have asked multiple unrelated questions in the same post. I have addressed the first one. Please…

Q: Use the definition of compactness to prove that a closed subset Y of a compact set X ⊆ R is compact

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Q: The cost equation for gizmos is Cost(q) = (94+ 5+ g) and the demand function is price(a)=1.4 Find…

A: From the given  cost equation and the demand function, We have to Find the maximum profit.

Q: Give the definition of a Lebesgue measurable set. Prove that if A and B are Lebesgue measurable…

A: We are given that A and B are Lebesgue measurable sets.        => m(A) < ∞,            m(B)…

Q: A and B are open subsets of X with Cl(A) = X = Cl(B).

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Q: Call a set A CR clopen if it is both open and closed. Prove that the only clopen sets are Ø and R.

A: A⊆R clopen if it is both open and closed.  We have to prove that the only clopen sets are ϕ and R

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: Prove the following. Let J be any set, and let {Xa : a E J} be an indexed family of connected…

A: It is given that Xα:α∈J is an indexed family of connected topological space that is each component…

Q: Let B be a set and let A ⊆ B be a subset. Suppose that there exists an injective function f : B → A.…

A: Schroder-Bernstein Theorem- Let A and B be arbitrary sets. Using the concept of cardinality. It is…

Q: Prove that a set is open iff it contains none of its boundary points; i.e., A is openAnA = 0.

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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: Let A and B be sets. Let X = A n B, Y = A \ X and Z = B\ X. Prove that X n Y = Ø, X n Z = Ø, Y n Z =…

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Q: 6. Prove that if x1, x2,..., Xn are distinct points in a Hausdorff space X. Prove that there are…

A: Let X be a Hausdorff space. Then in X any two distinct points are separated by disjoint open sets.…

Q: Let Y = {(x, y) = R² | x ‡ 1 and y ‡ 1} be a subset of R². Show that Y is an open set in the…

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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: жат fuss vodo у льа изба и и & Dis PAS

A: To prove that an open ball in a metric space is an open set. Using the definition: Consider set A…