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…

A:

Q: 4- A subset [a, b] R is compact set in: a- (R, Td), b- (R, T), c-Both (R, Tu) and (R, T₁) d- No one.…

A: As per our guidelines we are supposed to answer only first part. Kindly repost other parts in 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: Since every open subset of Rn can be written of countable union of disjoint open balls and each ball…

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: 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: 13. The interior of a set coincides with the interior of its closure..

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

Q: solve question (g),(h) with complete explanation asap

A: We use Hasse diagram and use definition of supremum and infimum.

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

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Q: Provide a direct proof that a sequentially compact set must be both closed and bounded without…

A: WE GIVE A DIRECT PROOF. First, we prove that set A is closed then A is bounded.

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.…

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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: Let ∼ ⊆ X ∗X be an equivalence relation on a set X. For x ∈ X define ̄x := {y ∈ X |x ∼ y}. Then it…

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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: Prove that any closed interval [s, t] can be written as the intersection of a collectable collection…

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Q: Give two different but equivalent definitions for a set CCR" to be closed, and prove they are…

A:

Q: Show

A: We prove this statement by using definition of Borel Set.

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

A:

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: -prove that for any bounded Set E, There is abounded set A That is a countable intersection fopen…

A: The solution is:

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: 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: 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?

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Q: Show that the only Hausdorff topology on a finite set is the discrete topol- ogy.

A: The question asks us to prove that the only possible Hausdorff topology on a finite set is the…

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 any convex open set is simply connected.

A: The objective is to prove that any convex open set is simply connected. Simply connected is nothing…

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: Prove that the unoin of a finite number of bounded sets is bounded.

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

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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…