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: Prove that in a topological space X, if U is open and C is closed, then U –- C is open and C – U is…

A:

Q: (1) Let X be an infinite set and set r to be the set given by T = {0} U {U C X : U° finite}. (a)…

A: τ={ϕ}∪{U⊂X :Uc is finite }  (a) Xc=ϕ, which is finite . so X∈τ let A,B∈τ Then Ac,Bcare finite…

Q: 5. If G1,G2 be two given closed sets in a metric space (X, d) prove that: G1 U G2 is closed.

A: We have to prove given result:

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

A: This is a problem of Topology.

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

A:

Q: Let X be a topological space. Show that the following are equivalent. (i) X is disconnected. (ii) 3…

A: Let X be a topological space then, To prove: Following statements are equivalent: (i) X is…

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

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Q: (b) Show that every compact set of T₂-space is closed set but the convers need not to true. (c) Let…

A: T2-space: A topological space X is said to be T2-space for given x, y∈X, there exists open sets U…

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…

A:

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: Suppose that X and Y are topological spaces and f: X- Y. Show that f is continuous iff for each…

A: Definition: Let X and Y are topological spaces and f : X→Y be a function. We say that the function f…

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

A:

Q: Prove: Let X and Y be two topological spaces. Then the function f:X →Y is continuous if and only if…

A: This is a problem of topology.

Q: Prove that any closed interval [s, t] can be written as the intersection of a collectable collection…

A:

Q: Theorem 4.4.8. Let (S, d) be a metric space. A set A is open if and only if its complement R\A is…

A: Given that a<b in ℝ The objective is to determine whether the given sets are open or not.

Q: let (x,t) be a topological space prove that (x,t) is not connected if and only if there exist A,B…

A: Let (X,t)  be  not  connected.We  have  to  show  that  X=A∪B ,where  A and B are  elements  in t…

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

A:

Q: 5. Let (X, d) be a metric space. i) Show that the open ball B, (ro) = {x € X : d (T, xo) < r} is an…

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

Q: 2. Construct two metric spaces X and Y which are homeomorphic, but X is complete and Y is not…

A: 2. Construct two metric spaces  and  which are homeomorphic, but  is complete and  is not complete.

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

A:

Q: The set [0, 1] with the discrete metric is compact. O True O False

A:

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: If X is a metric space with induced topology Ƭ, then (X,Ƭ) is Hausdorff. The contrapositive of this…

A: Given : (1) (ℝ, T) with the topology induced by taxicab metric.              (2) (ℝ, T) with the…

Q: Exercise. Prove that in a metric space (X, p), a set G C X is open if and only if it is a union of…

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Q: Hausdorff space.

A: Given: A set X with co-finite topology and  Hausdorff space. To Find: Under what condition a set X…

Q: Exercise. Prove that every finite subset of a metric space is closed set.

A: We have to Prove that every finite subset of a metric space is closed set.

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

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

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

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Q: 15. Let X be a normed linear space. Show that X is finite-dimensional if and only if every closed…

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Q: Suppose X is a topological space having only a finite number of components. Prove that each…

A: given that  suppose X is a topological space having only a finite number of components . we have to…

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

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Q: a) Consider the set X = {1,2,3} with the topology t= (0,X, (1). (2, 3}. (1, 2, 3} }. Show that (X,r)…

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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…

A:

Q: Let x be an infinite set equipped with the indiscrete topology T then (X,T) is compact. True or…

A: An infinite set equipped with discrete topology is not compact.