Theorem 1.10. A set is infinite if and only if there is an injection from the set into a proper subset of itself.

Q: 1

A: Consider the provided question, We have to prove by using only definition of a compact set, that (0,…

Q: Sketch a model that satisfies atleast 2 axioms of Fano's Geometry and Identify those axioms.

A: In this type of question, first we need to know, about Axioms of  Fano's Geometry, after this, we…

Q: i. PROVE THAT THE EXCLUDED POINT TOPOLOGY ON ANY SET WITH AT LEAST 2 POINTS IS NOT HAUSDORFF. ii.…

A: Definition: A topological space X, τ is said to be Hausdorff if for every x, y∈X there exits U, V∈τ…

Q: We say that a space (X, τ) is not connected if there exist two nonempty openings A, B such that A ∩…

A: Open Set: A X, τ be a topological space. Then the sets in τ are called the open sets of the…

Q: Ametric space (x,d) is complete if and only ifx has the cantor intersection property

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Q: Give an example of a set in R ^ 2 with the usual topology that is not connected but has a connected…

A: Let's examine the  problem for path-connectedness example of a set in R2 with the usual topology…

Q: Let (X, d) be a metric space with X being an infinite countable set. Show that X is not connected.

A: A set is said to be infinite countable if its elements can be counted like natural number but you…

Q: A ° = A – JA

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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: Exercise 2.46. Show that: (1) The topology generated by a subbasis is a topology. (2) The sets in…

A: 2.46

Q: Show that an infinite intersection of closed sets F, k = 1, 2, 3, . .., in a metric space (S, d) is…

A: first we will proof that an infinite union of Open sets in metric space ( S, d) is open using the…

Q: (b) Show that the continuous image of compact set is compact but the convers need not to be true.…

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Q: Select all that apply: The set II-1(-j,j) as a subset of X = R" is open in - The Discrete Topology…

A: In the given question we have to tell about the set s s.t why the set  is open in Discrete topology…

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: 3) Consider the subsets [0, 1] and [3,4] of R, the set of all real numbers with the euclidean…

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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: 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: Give example open set is compact

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Q: 2. Is there a set A that satisfies A = {A}? If yes, exhibit one %3D such. If not, Why not exactly?

A: No, A≠{A} for any set A. Let  A=1,2,3 then number of elements in A are 3. Now,  A=1,2,3 here the…

Q: show that (0.1)with usual topology is connected

A: Sol

Q: 3.1 Show that an infinite intersection of closed sets F, k = 1,2, 3, ..., in a metric (S, d) is a…

A: Let {Fk}  be an arbitrary collection of closed sets . . 1  i.e. for k = 1, 2, 3, . . . . . . . . To…

Q: Theorem 1.14. The set of all finite subsets of a countable set is countable.

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Q: Please define compact space and the clousure of .a set

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Q: Let X and Y be two topological spaces. X is connected and Y is disconnected are they homotopy…

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Q: Theorem 2.1. Let{U}{_, be a finite collection of open sets in a topological space (X,T). Then N, U…

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Q: Give an example of a closed set in R ^ 3 with the usual topology that is not compact

A: Given the usual topology on ℝ3. i.e. τusual=U⊆ℝ3: ∀x∈U, ∃∈>0 such that B∈x⊆U

Q: how to prove a map is unique?

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Q: Exercise 2.7. Give an example of a topological space and a collection of open sets in that…

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Q: Show that the set given in the image below is non convex. Please prove the result mathematically…

A: A set is said to be convex if for any two points in the set the line segment joining those points is…

Q: Hausdorff space.

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

Q: Use the definition of compact set to prove that the set [0,1] is compact

A: We know that a set E is compact if every open cover of E has a finite subcover. We suppose, that O…

Q: union of a collectionm of connected sets that have a point in common is conne

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Q: Consider the subsets [0, 1] and [3, 4] of R, the set of all real numbers with the euclidean…

A: Consider the subsets [0, 1] and [3, 4] of R, the set of all real numbers with theeuclidean topology.…

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

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Q: Prove that a nonempty set is bounded if and only if it is contained in a bounded interval

A: A non empty set S is called bounded if it has both upper and lower bounds, In a certain sense a non…

Q: every finite subset of metric space is compact. ture or false?

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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 any subset of bounded set is bounded

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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: 4) Consider three-dimensional Euclidcan space R with the usual Euclidean distance, definition of…

A: This is a problem of non-euclidean geometry, euclidian postulate.

Q: Show that all open balls are bounded subsets of a metric space.

A: Solution: Let M=X,d be a metric space. For an arbitrary point a∈X, the open ball Bεa where ε>0 is…

Q: Theorem 1.12. The union of countably many countable sets is countable.

A: We have to prove that the union of countably many countable sets is countable.

Q: If fis a bijection, find the sets X and Y.

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Q: Consider the subsets (0, 1) and [2, 3) of R, the set of all real numbers with the euclidean…

A: Consider the given sets. (0, 1) and [2, 3) We have to show that (0, 1) is not homeomorphic to [2,…

Q: Theorem 1.11. The union of two countable sets is countable.

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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: 3.1 Show that an infinite intersection of closed sets F, k 1,2, 3,..., in a metric snace (S, d) is a…

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Q: Example 2.17. On N, let {{0,1}, {2,3}, {4, 5},..} be a subset of a topology Tad. What other sets…

A: Suppose there exist a set S which has total number of subset equal to T. The collection of all…

Q: Theorem 1.13. The set Q is countable.

A: