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

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A:

Q: Let F = {Fα : α ∈ A} be a family of compact subsets of R. Prove that an intersection, α∈A, Fα is…

A: A set F is said to be a compact set if it is closed and bounded both.

Q: Let Y be an infinite subset of a compact set X c R. Prove Y' # Ø.

A: Y' denotes the derived set of Y i.e. the set of all limit points of Y. We just need to show that any…

Q: be a closed subset of space Xa

A:

Q: Let E1, E2, ..., En ⊂ X,d be compact subsets of a metric space X. Just using the definition of…

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

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Q: A map f:x-y is said to be an open map if for every open set U of X, the set f(u) is open in Y. Show…

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A:

Q: Prove if {Galpha}alpha e A is a collection of closed sets in a metric space, then…

A: Recall that arbitrary union of open sets in X is open in X.

Q: For a normed space X and a closed convex subset A of X. Prove that (º )nÎ= Â. A

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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: Let (X,7) be a topological space and ACX. Show that (n) A is open if and only if Int(A) = A. (b) A…

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Q: Let A be a subset of a topological space X. Show that bd(A) = Ø iff A is open and closed in X.

A: We need to prove the statement.

Q: Let X be a discrete spaces then O X is homeomorphic to R if and. only if X is countable O X is…

A: I have provided a solution in step2

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

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Q: Prove that T is the discrete topology for X iff every point in X is an open set

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Q: let C be a compact subset Tn a Haus dorft topolopfcal space (K,R) and bex a If bac then prove that…

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Q: Question 10. Prove or disprove. Any infinite subset A of a discrete topological space (X,r) is…

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Q: For any infinite set X, the co-countable topology on X is defined to consist of all U in X so that…

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

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

Q: Let X = {a,b,c} prove that topological space on the set X T = {X. Ø, {b}, {a,b}} be a

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A: The objective is to choose the correct option: Let X be a discrete space then  a) X is homeomorphic…

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…

A:

Q: 4. Let (X, d) be a metric space and xnx where each x, e X and x e X. Let A be the subset of X which…

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Q: Let X be an infinite set endowed with the finite complement topology T, then (X,T) is compact. * O…

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Q: Let {An : n e N} be the indexed collection of sets defined by (2. 2-) 1 An Prove that: = Ø

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Q: Is the following statement True or False? Justify each answer. If S is a closed bounded subset of a…

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Q: If x is a limit point of a subset A of a metric space X, then sh

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Q: 2. Let X be any uncountable set, Prove that: 7. = (GC X: G is countable} U {0} is a topology on X.

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Q: Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set.

A:

Q: (1) Determine if each of the following statements is true(T) or false(F). 1- In a metric space,…

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Q: Let A CX with the discrete topology. Prove that ô(A) = Ø.

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

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Q: Suppose p: X → Y is a surjective map from a topological space X to a set Y. Verify that the quotient…

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Q: .2 Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set.

A:

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

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

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Q: Let (X, d) be a metric space and let A be a non-empty subset of X. Prove that A is open if and only…

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Q: Let (X7) be a topological Space and Show that A is open and closed m (xx) iff %3D

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Q: f A∩B≠∅ and A∩Bc≠∅ then show that A∩X\B̅≠∅ and A∩B̊≠∅

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Q: Let X = {1, 2, 3, 4, 5, 6}. Prove that in general, if X is finite there is only one topology for X…

A: Let X = {1, 2, 3, 4, 5, 6}. Prove that in general, if X is finite there is only one topology for X…

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

A: False

Q: If A is a compact subset of a metric space (X, d) and B is a closed subset of A, prove that B is…

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Q: Let (X,T) be a comapct space and A be a subset of X. If A is closed, then A is compact. O True O…

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Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.