. Let X be a topological space, and let A be a subset of X which is both op closed. Show that A is a union of connected components of X.

Q: 3. Let (X, 7) be a Hausdorff space. If X is finite, then show that 7 = P(X), where P(X) is a power…

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Q: (E, Q4: state and prove L is a normed space ? e a ne. Vila a convex set.

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Q: Theorem 8.6. Let C be a connected subset of the topological space X. If D is a subset of X such that…

A: Let C be a connected subset of the topological space X.Also given that D is a subset of X such that…

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: Theorem 2.30. Let A be a subset of the topological space X, and let p be a point in X. If the set…

A: Given that A is a subset of the topological space X and p is a point in X. Consider a sequence…

Q: Show that if a topological space has a finite number of points each of which is closed then it has…

A: Topology question.

Q: (iv) Prove that for any sets A, B,C we have A U (B nc) = (AU B) n (AUC)

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

A: it is given that X is a topological space which is locally path connected and U be an open set in X.…

Q: Exercise 3.45. Let X be any space and Y be a discrete space. Show that the Cartesian oduct X X Y is…

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Q: Let S be the subset of the ℝ2 given by all pairs (x, y) so that x2 + y2 = 2. Show that S is a closed…

A: Let, s be the circle given by, s=x,y∈ℝ2 :x2+y2=2  Then s is a closed subset of ℝ2. Let us consider a…