3. A space is totally disconnected if and only if the only connected sets are the sets of a single point. Prove that if X has the discrete topology then it is totally disconnected, is the converse implication true?

Q: 14) A subset of topological space X is dense in X iffA a) intersect every set c) intersect every…

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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: Let U = R be the universal set and A = [1,3], B = (-∞,6], C = (-4,0). a) AUB b) BnC c) A-C d) BC

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Q: We say that a space (X, τ) is not connected if there exist two nonempty openings A, B such that A ∩…

A: Open Set: A X, τ be a topological space. Then the sets in τ are called the open sets of the…

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: 6. Show that in a topological space, the union of finitely many compact subsets is com- pact. Is the…

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Q: 9. The set {0} in the indiscrete space X is A.open but not closed B.closed but not open C.both open…

A: In the indiscrete topology  open sets are empty set and set x itself. Hence the closed set are empty…

Q: 4. Prove that if A and B are convex sets, then AN B is also convex.

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Q: 4. Let A and B be two closed subsets of the connected topological space X. Suppose that X = AUB And…

A: Method of contradiction : suppose A is disconnected ,then there are two disjoint closed non empty…

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: 17. Prove: S' is a closed set.

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Q: 7) Let A and B are subset of a space X such that ACB then a) If B compact set then A compact set. b)…

A: Let  and  are subset of a space  such that , thena) If B is compact set, then A is compact set.b) If…

Q: Let T be the finite closed topology on a finite set X, then * O Tis the Euclidean topology O Tis the…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: 6. A relation R on a set X is called a total order if it is antisymmetric, reflexive, transi- tive,…

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Q: We say that a space (X,τ) is connected if there exist two open non-empty spaces A,B such that A∩B≠∅…

A: Let X,τ be a connected metric space. We have to show that the only subset of X are both open and…

Q: 2) Show that the digital line topology on Z is not Hausdorff.

A: Given that Digital line topology on ℤ, we have to prove that it not Hausdorff Therefore, Let a space…

Q: Exercise 2. Suppose X is a topological space and A, B are subsets of X. a) Show that A\BCA\B. b)…

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Q: Show that there is no topology on X={a,b,c,d,e} based on the family B={{a,b},{a,b,d},{b,d,e}}

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Q: 4. Let X be any set and 7 be the countable complement topology on X. b) If X is uncountable, then…

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Q: topological space a

A: Presumably, the closure of Q in R under the usual topology. It suffices to show that for every real…

Q: 2. Show that totally disconnected is a topological property.

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Q: Let (M, d) be a compact metric space. Show that closed subsets of M are compact.

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Q: 3. Let A = {a} be a bounded subset of R. Which of these three statements about A is/are not true? I:…

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Q: Problem 3. Let X be a finite set. Prove that the finite complement topology on X is the discrete…

A: Note: The finite complement topology is also called cofinite topology. And the discrete topology is…

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

A: We use definition of connected and path connected sets. We use contradiction method.

Q: if

A: R is not connected if T is the finite closed topology.

Q: A, B be subsets of a metric space. Show that A U B = B and that A B. Give an example to show that An…

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Q: For an indexed collection of sets to have an intersection an element needs to be in every set right?…

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Q: 1) Let f: A ->B with C1 C C2C A and Dic D,CB Which is true? a) f(C) c f(C2) OR b) f(C2) c f(C:) (i)…

A: Given: i) To prove that answer is correct. ii) To give a counterexample to show that the set…

Q: 5. Describe the closed sets in the following topological spaces: a) (X, E,) wherepe X. b) (X, P,)…

A: (a) Given,           (X, Ep), where p∈X

Q: 11. Show that the set of real numbers with the usual topology is not homeomorphic to the R set of…

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Q: Are the following true or false? If true, prove. If false, give a counterexample. (a) The…

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Q: ☺ Let (u,Ue...,un) be a lineorly ndependent list. Which True Statement? is a has --- a non-zero…

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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: Show that if a topological space has a finite number of points each of which is closed then it has…

A: Topology question.

Q: 3) Let D = {(x, y) = R² | 0 < x² + y² ≤ 4}. Give all boundary points of D. Is D open, closed or…

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Q: 7. Show that the Obenchain class LOb(B) is a convex set.

A: Given: To show the Obenchain class LOb (β) is a convex set is given as,

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: 3) let X be discrete topology and Ac X, find A ?

A: