Question No 1 (8). Prove that the discrete topology on an uncountable set does not Satisty the Second exiom Solution:- ot Countability-

Q: Show in the discrete metric space: 1. All sets are open (especially single points);

A: Let X be a non-empty set and d be discrete metric on X

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: In the discrete topology on the set X, every set is open and it also follows that every set is…

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Q: Set A is bounded above if and only if set -A is bounded below.

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Q: 4. (a) If X is an infinite dimensional space, show that the set {x = x | || x || = 1} is not…

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

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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…

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Q: Let (X, E, µ) be a measure space. Justify through an example that the hypothesis µ(X) < o in…

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

A: 2.46

Q: Prove or disprove. A closed and bounded subset of e' is compact.

A: PROVE OR DISPROVE :-  A closed and bounded subset of l1 is compact.

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: let x be an infinite set and let ta be a topology on x in which all infinite subsets of x are open…

A: Given that X be an infinite set and let τ be a topology on X in which all infinite subsets of X are…

Q: Show in the discrete metric space: 1. All sets are open (especially single points); 2. All…

A: We first define Discrete space

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: 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: 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: Suppose metric space X is not path-wise connected then X is not connected. O True O False

A: Question 9 Answer: as a path connected space of employees connected but converse is not true. So I…

Q: m) Show that if T;,T; and Ty are topologies on a set X.then r, nnt, is a topology on X.

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

Q: Make the definition of tupological non-empty set and * E X. Inves tigale whether Space Let X be…

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Q: 6 Prove that a closed set in the metric space (S, d) cither is nowhere dense in S or clse contains…

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Q: 4.5 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

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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: Is the Cantor set closed? Is it compact? Explain!

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Q: rcise: Prove that every finite set S of R" has no accumulation point.

A: Proof of the the given theorem is given as

Q: the collection of all close subsets of R is called the topology ofR ylgn İhi

A: the collection of all close subsets of R is called the topology of R

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

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Q: If X #0, compare the following topologies on X: a) The co-finite topology 7, and the co-countable…

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Q: Theorem 2.30. Let A be a subset of the topological space X, and let p be a point in X. If the set…

A: Given that A is a subset of the topological space X and p is a point in X. Consider a sequence…

Q: Hausdorff space.

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

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

A: The given is related to Metric spaces. The solution is explained below.

Q: X is not open, must it be closed? Either prove that every subset E that is not open must be closed,…

A: We will show it by giving example

Q: Show that any compact set in R is closed and bounded.

A: Proof

Q: Let X be an infinite set with the finite closed topology T={S subset of X; X-S is finite}. Then

A: a) If not, then there are non-empty disjoint open sets  such that  . Since they are disjoint, so  .…

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: Set A is bounded below if and only if set -A is bounded above. True False

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Q: Lecture 8 Review Exercise: 1) What does it mean if we say that a set S is closed under a binary…

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Q: (a) If {Ta} is a family of topologies on X, show that N Ta is a topology on X. Is UTa a topology on…

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

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: Question 2 Prove directly (i.e. from the definition of compactness) that if K is a com- pact subset…

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

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Q: Let X = {a, b, c}. Is the collection {X, Ø, {a}, {a, b}, {c}} a topology on X?

A: Here, the given set is X=a,b,c and collection is X,∅,a,a,b,c.

Q: Consider X = {a1, a2, ..., an} Can a topology of X with 3 open and one with 4 open be equivalent?

A: We have to verify the given statement.

Q: Prove that every set with positive Lebesgue measure contain a non-measurable set

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Q: An indexed family of sets (A,hcs is said to be a cover of a set X if and only if X CU,crA Use this…

A: Let A be a non-empty set and Cii∈I and Djj∈J are two different covers of A. Since Cii∈I is cover of…

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: Let T be the finite closed topology and be also the discrete topology on a non empty set X. Then* OX…

A: Gievn that Let T be the finite closed topology and be also the discrete topology on a non empty set…