9. Show that the property of "having nonempty open sets with a finite number of points" is a topological property.

Q: 13. Let X and Y be subsets of any finite, nonempty set. Give the properties of the relation where X…

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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. 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: Let Gbe an open set in R". Two points x, Y e G are said to be equivalent if they can be joined by a…

A: Since every open subset of Rn can be written of countable union of disjoint open balls and each ball…

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: 6. Show that the span of this set is all of R?. {}}

A: Let S=v1,v2,...,vn be a set of vectors in a vector space V(F) . The vector space V is spanned by S,…

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: 37. Which of these three statements is/are not true? I: Every finite nonempty subset of R has a…

A: We shall answer all the parts and provide justification as to why a particular part is not true. In…

Q: Show that relint (C) is nonempty for any nonempty convex set CC Rª.

A: Relative interior of C : Let C be a convex set. We define the relative interior of C as the set of…

Q: . Is it true that the null set is a subset of every other set? (Y/N)

A: If a set has no element then the set is called null set. Null set is also called empty set. It is…

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: Use graph to find  set. [1, 3) ∩ (0, 4)

A: Given set is  [1, 3) ∩ (0, 4) The graph of the given set is as shown

Q: Identify the type of set (finite, infinite or empty) as described by the following: 1. A = {stars in…

A: A set is said to be finite if it contains only finite number of elements. A set is said to be…

Q: Explain with an example of a topological space which is singleton that is open but not close and…

A: Example in refer of not  open but closed .Take , Here A is singletone set which is closed but not…

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: Show by example that the union of two convex sets need not to be convex.

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Q: Give an example of a bounded set that has infinitely many isolated points

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Q: Find a topology in which the point x = 5 is an open set.

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

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Q: What is the minimum number of points that can be contained in a space

A: There is no minimum number of points to be contained in a space. But if we want to find a location…

Q: 4. Which of the intervals (-3, 3), [4, 7], (-4, 5], (0, ∞) and [0, ∞) are open, closed, both, or…

A: given some intervals to identify whether open, closed, both, neither

Q: How to Apply the Archimedes'principle to prove that a set is empty

A: Consider the given information. the Archimedes' principle is defined as, Property states that for…

Q: 5. Show that a finite set is a closed set.

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

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

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: 4. Which of the following pairs of sets are equivalent or equal? (a) A = {x : x € N, x < 6} B = {x :…

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Q: How do we denote the empty or null set?

A: A set is said to be an empty set or a null set if there is no element in it.