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: Show that any totally bounded subset in a metric space is bounded.

A: Let A be a Totally Bounded Set Totally Bounded = For Each ∈ > 0, there exists an ∈-net. Let ∈>…

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

A:

Q: are closed.

A:

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

A:

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

A:

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: 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: Exercise 1. Give examples of subsets A, B in a topological space X (for example X = R or X = R²)…

A: To give an example of subsets A,B in a topological space X such that both A,B are connected but A∪B…

Q: Show that any finite union of Nowhere Dense Sets is Nowhere Dense.

A:

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

A:

Q: Prove that closed balls are closed sets in the standard topology on R?.

A: Topology Question

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…

A:

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

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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: Theorem 2. Let R and S be binary relations on a set X. Then, (a) (R-1)-1 = R. (b) (RUS)-1 = R-'Us-!.…

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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: How can I show that the collection  A composed of empty set and all cofinite subsets of X Is a  sets…

A: