a)Every subset of a topological space is either open or closed. b)If X is a topological space with the discrete topology and if X has least two elements, then X is not connected. c) True or False: If X is a topological space, then there always is a metric on X which gives rise to its topology. d) True or False: If X and Y are topological spaces and if f : X → Y is a constant map (which means that there is a point y ∈ Y such that f(x) = y for all x ∈ X), then f is continuous. e) True or False: If X is a topological space, then X is both open and

Q: A topological space is Hausdorff iff limits of all nets in it are unique

A: This is a problem of Topology.

Q: 4. (a) If X is an infinite dimensional space, show that the set {x = x | || x || = 1} is not…

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Q: Every totally  bounded metric space is compact

A: This statement is not always true.

Q: State whether the following is true or false: a) The set [0,4] with the usual metric is…

A: (a) The set [0,4] with the usual metric is sequentially compact. This statement is true. Considering…

Q: that the boundary of a closed set is nowhere dense in

A: Matric space

Q: 7. Let X be a connected space. Let C be a connected subset of (X, T). If X\C=BUD where B and D are…

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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: 5. State whether each of the following statements is true or false, in either case substan- tiate/…

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Q: 3- A discrete topological space (X,T) is: a-To-space, b-T₁-space, c- T₂-space, d- No one.

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Q: Q6) Find a topological space and a compact subset whose closure is not compact. Note hat a space is…

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Q: (a) For the following subsets in an indicated metrie space detennine the interior, boundary points…

A: See the attachment.

Q: One of the following statements is true:

A:

Q: If X is a normed space then the open ball and closed ball is a Example: convex set

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Q: C. Show that a set A in R² is open in the Euclidean metric it is open in the max metric.

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Q: If X is a Hausdorff topological space, is the quotient space X/∼ necessarily Hausdorff? Explain.

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Q: 5. State whether each of the following statements is true or false, in either case substan- tiate/…

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Q: {X, Ø, {1}, {2},{1,2},{1,3}}, is (X, t) a connected a) Let X = {1,2,3} with the topology t=…

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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 A be a set and d : A × A → {0, 1, 2} a symmetric set function that satisfies d(x, y) = 0 ⇔ x =…

A: To prove that (A,d) is a metric space, we need to show that it satisfies the following properties:…

Q: Show that Rsta is Hausdorff.

A: Note that a topological space X,I is said to be Hausdorff if (and only if) for every distinct points…

Q: Suppose X is a connected topological space with the property that every point x of X has a…

A: In this problem, we consider a connected topological space X with the property that every point in X…

Q: Theorem 2.25. Suppose that (X, p) is a metric space. (a) The intersection of any collection of…

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Q: 3) (i) Let X be a set containing elements a and b. Define T= {X, Ø, U| a E U and b E U}. This is a…

A: Definition:  A collection of subsets I of X is said to be a topology if it satisfies following…

Q: Exercise 1. Let X be a metric space. The subsets of X which are arbitrary (finite or infinite)…

A: What is Open Set: Open sets in a metric space are those that, for every point P, include all other…

Q: X is not open, must it be closed? Either prove that every subset E that is not open must be closed,…

A: We will show it by giving example

Q: Prove that closed balls are closed sets in the standard topology on R?.

A: Topology Question

Q: 1. Two metrics d₁ and d₂ on the same set X are said to be topologically equivalent if they generate…

A: The data given in this question are two metrics on the same set X, where X is the set of all ordered…

Q: A, B be subsets of a metric space. Show that A U B = B and that A B. Give an example to show that An…

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Q: 1. prove every 2nd countable space is Lindelof space. 2. Every Lindelof topological space induced by…

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Q: Theorem 8.9. Let f : X Y be a continuous, surjective function. If X is connected, then Y is…

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Q: Find all limit points of a set B = {1, 1 2 , 1 3 , 1 4 , . . .} on the real line with the usual…

A: The set B is B=1, 12, 13, 14, . . . . To Do: We have to find the limit point of set B on the real…

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: 10. Let X and Y be topological spaces. Show that if either X or Y is con- tractible, then every…

A: Given X and Y are two topological spaces.

Q: Theorem 2.15. Let (X,T) be a topological space, and let U be an open set and A a closed subset of X.…

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Q: State whether each of the following statements is true or false, in either case substan- tiate/…

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Q: Every closed subset of a connected metric space is connected. O True O False

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

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Q: 6. Let (X, d) be a metric space and Y CX be nonempty. Verify that (Y,d) is a metric space. Show that…

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Q: QUESTION 3 Any cover of a metric space is open. O True O False

A: