Let X and Y be two discrete spaces, then * O X is never homeomorphic to Y X is homomorphic to Y if and only if X and Y are both infinite X is homeomorphic to Y if and only if X and Y have the same cardinality X is homeomorphic to Y if and only if X and Y are both 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:

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Q: Congruent is. Relation on Z Equivalence. anti symmetric anti reflexive Equivalence partial order

A: To solve the following problem by using the definition of equivalence relation

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A: It is given in part (a) that (X,t) be any topological space, and the interval (x-A,x+A) is only open…

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A:

Q: (6) If a space X is connected space, (And for any A, B are open sets, and A # # B), itt: (a) AUB if…

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Q: 10. Prove that for subsets X, Y, and Z of a given universal set U (a) X-Y=X-(XnY)= (XUY) - Y (b)…

A: “Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

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Q: R.4. Prove that the set of real numbers in the interval (0, 1) which have a terminating decimal…

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Q: (a) Suppose S is a subset of positive integers, ZZ. If |S| ≥ 3, prove there exist distinct z and y…

A: (a) Given that S is subset of  such that Observed that any positive integer is either even or odd…

Q: Let X and Y be two discrete spaces, then! X is never homeomorphic to Y X is homeomorphic to Y if and…

A:

Q: Every bounded domain subset of D containing

A: Set theory

Q: Which one of the following statements is true? * R with the Euclidean topology and R with the…

A: Two homeomorphic spaces share the same Topological property If one of them is compact, then the…

Q: 2. Show that the following i two sets have the same |Y|≤|X: cardinality, by proving X ≤ Y and X = R,…

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Q: Prove or disprove If P is ordered set and f is an order preserving then P is isomorphic to f(P)

A: The statement "If P is an ordered set and f is an order-preserving function, then P is isomorphic to…

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 and Y be two discrete spaces, then * X is never homeomorphic to Y X is homomorphic to Y if and…

A:

Q: Given two sets X and Y, we denote the cardinality of X by |X| and the set of all functions from X to…

A: We are given two sets X and Y. The set of all functions from X to Y is denoted by YX.  Suppose that…

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A: To Prove: x,y,≤ is not well ordered, where x,y is an open set with the usual order "≤".

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A: Topology question.

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A:

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…

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A:

Q: Let T be the finite closed topology and be also the discrete topology on a non empty set X. Then…

A: X is finite

Q: Let X be a set with an operation *:X² → X such that (i) x * x = for all x E X, %3D (ii) x * (y * z)…

A:

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

A: Suppose that X is locally connected. Let U be an open set in X and let C be a component of U. If x…

Q: Let A be a non-empty funute set and let S = P(A). Define relation R ⊆ S × S by: xRy ⇔ |x| = |y|,…

A: It is given that S=PA, where A is a set and PA denotes the power set of A. The relation is defined…