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

Q: Write true otherwise false if the pairs of sets are equal,{{1}},{1,{1}}

A: Two sets A and B are equal if A and B  have the same elements.

Q: i. Provide an example to prove that the arbitrary intersection of open sets is not an open set ii.…

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Q: Give an example of a set with a finite number of points that is not se-quentially compact, or else…

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Q: Prove a neighborhood is an open set

A: Recall that, a neighbourhood of a point is a set of points containing that point where one can move…

Q: Provide a counter example that infinity intersection of open set is open

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Q: For each condition below, give an example of a set that satisfies that condition, or prove that one…

A: The given of the question is a list of conditions related to sets and subsets of real numbers, and…

Q: Define Closed sets ?

A: Closed sets are defined as a set which consists all of their limits points. It is also defined as…

Q: Prove that (2,5) is an open set

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Q: A topological space is dis connected if and only if it contain non empty set which is both open and…

A: I give you two simple example with explanation

Q: (a) Let A, B be sets. Prove that A\(An B) = A\B. You may assume all sets in the question are…

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Q: Give an example of a set that contains nine elements of which two are sets. List three Subsets of…

A: Let the set which contains 9 elements be, A=1,2,3,4,5,6,7,8,9

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: Determine whether the following statements are true or false: (a) The empty set ∅ is a subset of {1,…

A: Empty set is a subset of every set. A Power Set is a set of all the subsets of a set.

Q: List all the possible subsets of set G, where G = {-7,m,z}- %3D

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Q: Can a topology on an infinite set not be discrete and indiscrete in which every subset is clopen?

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Q: >) Recall that the symmetric difference of two sets X and Y, denoted XAY = (X-Y)U(Y-X), is the set…

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Q: Prove that derived set of a bounded set is also bounded.

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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: Let A be a partially ordered set that has the least upper bound property. Then every nonempty subset…

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Q: Consider two sets A and B such that A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}. Find the set C…

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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: a and b be disjoint subsets of the universal set U. Assume n(U) = 110, n(A)=50 and n(B) = 25. Find…

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Q: Prove that a finite union and an arbitrary intersection of compact sets is a compact set.

A: Step 1: Step 2: Step 3: Step 4:

Q: show that empty set, X is an element of S intersection T in topology

A: Here We have to show that empty set must be in S∩T where S and T are topology on set X.

Q: VII.1. Describe the set (R × Z) ∩ (Z × R)

A: We have to describe the set (R × Z) ∩ (Z × R).

Q: Let X be a nonempty set with subsets, A, B, and C. Prove that A ∩ (B \ C) = (A ∩ B) \ (A ∩ B).

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Q: Use graph to find set. [2, ∞)∩ (4, ∞)

A: Use the graph to find [2,∞)∩(4,∞)

Q: 4. Prove that the set Z is a closed set in R.

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Q: is the set [5,100] compact?

A: Given that:A set is S=5,100.We have to check whether a set 5,100 is compact or not?

Q: Give an example of two sets A and B which are homeomorphic, A is complete and B is not.

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Q: Give example closed set and bounded set is compact

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Q: I) We know that {-5,-4,-3,-2,-14 and IN are well-orderec sets with respect to the trivial ordering.…

A: Given information :  -5, -4, -3, -2, -1 and ℕare well ordered sets with respect to trivial ordering.…

Q: Determine whether or not the given set is (a) open,(b) connected , and (c) simply-connected. {(x,y)…

A: Determine whether or not the given set is (a) open,(b) connected , and (c) simply-connected. {(x,y)…