Let X be a topological space. Prove that (a) Ø and X are closed. (b) Intersections of any number of closed sets are closed (c) The union of two closed sets in closed.

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…

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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: A metric space (X , d) and a collection of open sets U such that UEU is not open.

A: Let X, d be a metric space. Let U be the collection of open sets in the metric space X, d. That is,…

Q: Prove that the quotient of a locally connected topological space is locally connected.

A: We have to prove that Let q:X→Y be a quotient map, X is a locally connected space. Show that xY is…

Q: Let (X,T) be a separable topological space, let A be an open subset of X. Then (A, T,) is a…

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Q: Let (X, I) be a topological space and suppose that A and B are subsets of X such that ACB. Prove…

A: Let x ∈A°. Then by definition, there exists an open set U  such that  x∈U⊆A.Since A⊆B hence x∈U⊆B ,…

Q: the collection of all close subsets of R is called the topology of R

A: To check: Whether the statement is true or false: Given: The statement is "The collection of all…

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

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

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Q: (c) Prove that a metric space is connected iff it contains exactly two sets that are both open and…

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Q: Use the Heine-Borel theorem to give an example of a set not compact in R with the usual topology…

A: An example of a set not compact in R with the usual topology and a non-compact set R2 with the usual…

Q: Let (X,T) be a topological space Property C=P.C.  A subset A of x has P.C If it's subset of the…

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

A: In this question, we use the definition of compactness to prove that a closed subset Y of a compact…

Q: Let X and Y be two locally compact spaces, then X*Y is locally compact. (Recall that: X is said to…

A: This is a problem of topology.

Q: Let

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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: Let (Ka)aeA be a collection of compact subsets of a metric space, X. Show naeAKa is closed in X.

A: A metric space X is compact if every open cover of X has a finite subcover.

Q: Let (X, T) be a topological space and A C X. Prove that A =A.

A: PROOF :

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: Exercise 2.27. Show that a set U is open in a topological space X if and only if every point of U is…

A: Let, U is an open set in a topological space X. Then, U⊂U  So, U⊂int(U) and int(U)⊂U Hence, U=int(U)

Q: Let (X, t) be a topological space and A, B EX, then 1- (X- A)° CX-Ā 2- (X-A)° 2X-Ā 3- (X- A)° = X-A…

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Q: Let X And Y be two distrecte spaces. Then X is homeomorphic to Y if and only if X and Y are both…

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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: Let (X, T) be a topological space, AcX, then A'next (A) = 10 AO

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Q: Provide an example of a topological space that is connected but not path-connected,why?

A: We use definition of connected and path connected sets. We use contradiction method.

Q: let C be a compact subset in a Hausdorff topological space (X,T) and b∈X a point. If b∉C then prove…

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Q: Prove: If F is a closed subset of a compact space X, then F is also compact.

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

A: Topology Question

Q: Show that any compact set in R is closed and bounded.

A: Proof

Q: 5. Describe the closed sets in the following topological spaces: a) (X, E,) wherepe X. b) (X, P,)…

A: (a) Given,           (X, Ep), where p∈X

Q: Let X be an infinite set with the finite closed topology T={S subset of X; X-S is finite}. Then

A: a) If not, then there are non-empty disjoint open sets  such that  . Since they are disjoint, so  .…

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

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Q: Let X be a set equipped with the finite complement topology show X is compact Show that X is…

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

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Q: (a) Prove that every nonempty proper subset of a connected space has a nonempty boundary. Is the…

A: As per Bartleby guidelines, for more than one question asked, only first is to be answered. Please…

Q: Let X be a topological space with the property that, for every topological space Y, every function f…

A: The space X with the discrete topology means every singleton of space is open.

Q: Let X be a discrete spaces then * X is homeomorphic to R if and only if X is finite X is…

A: Given that, X is a discrete space. To be homeomorphic X has to be of same cardinality as ℝ and so X…

Q: Let X be a topological space and A be a subset of X. Prove that (A°)° = Ac and (A) = (A°)°.

A: Let A be a subset of a topological space X. Let the interior of A be denoted by A∘. Let x∈A∘c. Then…

Q: Prove that in a metric space (X, d) every closed ball that is a set K(x, r) = {y e X : d(x, y) <=…

A: Question: Prove that in a metric space X, d every closed ball K¯x, r=y∈X, dx, y≤r is a closed set.…

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…

Q: (a) Show that every metric space is T2-space.

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Q: let T be the finite closed topology on any set X.Every subset of (X,T) is connected true or false?

A: let T be the finite closed topology on any set X.Every subset of (X,T) is connected true or false?

Q: Exercise 60 Let p and q be two points in a topological space X. Show that there exists an open set U…

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Q: Theorem 2.20. For any set A in a topological space X, the closure of A equals the inter- section of…

A: Let A  be a set in the topological space X. To prove that the closure of A  is the intersection of…

Q: Show that a space X is Hausdorff if and only if the diagonal {(x, x)|x € X} is closed in X2.

A: We use basic topological definition to prove this.

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

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Q: Let X be infinite set and x0 ∈ X. Consider the topology τ = {G ⊆ X : x0 ∈ G or G = ∅} on X. Is (X, τ…

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

A: Solution