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: If  X is a countable set show that every topology on X is second countable.

A: To Show: If  X is a countable set show that every topology on X is second countable.

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: 11. If a set has empty interior, then so does its closure. 12. The closure of a set coincides with…

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Q: Theorem 1.10. A set is infinite if and only if there is an injection from the set into a proper…

A:

Q: Is the relation of “mapping” between two sets used to determine cardinality - and not used to…

A: Note- Since you have asked multiple question, we will solve the first question for you. If you want…

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: 2- Every finite set in any topological space is a- Regular set, b- connected set, c- Hausdroff' set,…

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Q: 3.1 Show that a finite intersection of closed sets is closed.

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

A:

Q: 1. (a) For a set A, P (A) is the set of subsets of A. Write down all the different elements of the…

A: (a)  The empty set, ϕ  is the subset of any set and a set is always a subset of itself. P(ϕ)={ϕ},…

Q: Show that compact sets are closed under arbitrary intersections and finite unions. (Here we're…

A: To show that compact sets are closed under arbitrary intersections and finite unions

Q: Prove that in R with the usual topology, the only open sets connected by paths are the open…

A: the solution is given below

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: 1. a) Suppose A, B and C are sets. What do you need to show to prove the following using the…

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Q: Prove that if A is transitive set, then A+ is also a transitive set.

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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: Let ∅ be the empty set. Let X be the real line, the entire plane, or, in the three-dimensional case,…

A: Given:

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 relation ''<'' is 1) İrreflexive. 2) Transitive. called a quasi-order on a set S if its: صواب ihi

A: We know that

Q: 4. Prove that the intersection of an arbitrary collection of compact sets in R is compact.

A: (Heine-Borel Theorem). A subset A of R is compact if and only if A is bounded and closed (with…

Q: Q.Prove If ANB is pathwise connected set, then A andB are or disprove the following: pathwise…

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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: 1) Let the set S be defined as S = {x | x is a U.S. President} Find n(S). 2) Define compliment to a…

A: use the basic property of sets .

Q: Prove that: (i) A is closed. (ii) A = nc. ก ACC closed (iii) A is the smallest closed set in X that…

A: Given the set A which is the closure of the set A, i.e. a set that contains all point of A as well…

Q: Let A and B be open sets in a topological space. Is A ∩ B open?

A:

Q: 1. (a) For a set A, P (A) is the set of subsets of A. Write down all the different elements of the…

A: Note: We are entitled to solve only the first question, unless specified 1) (a) We know the power…

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

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Q: Proposition 1.35. The intersection of any family of closed sets in a metric space X is also closed…

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Q: Broadly speaking, what is a mathematical closure? O A. Only relations have closures. They are…

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Q: Prove that every Union of Open sets is also Open.

A: To prove: Every Union of Open sets is also Open.

Q: If D(L AOB) states the set of points located on the OA side containing point B and the OB side…

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Q: Theorem. Show that Closed subsets of compact sets are compact.  Need handwriting solution only.

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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: 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…

A:

Q: Prove that every nonempty subset of a well-ordered set is well-ordered.

A: Given problem is :

Q: Topology:Q13 or 14 For each of the following, if the statement about a topological space is always…

A: Solution (13):-" the interior of the closure of a set equal to the interior of that set"    The…

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: We know that Since every open subset of Rn can be written of countable union of disjoint open balls…

Q: 4. If X #0, compare the following topologies on X: a) The co-finite topology Tf and the co-countable…

A:

Q: what ordered pair is an example of a topological space that is connected but not path-connected,why?

A: We use contradiction method.

Q: Prove directly that the union of two disjoint sets, where one is countably infinite and the other is…

A: