6. A space X has the property of being separable if it contains a countable dense set (its closure is the whole space). Show that the property of being separable is a topological property.

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

A:

Q: Exercise 3.55. Show that every wedge sum of Hausdorff spaces is Hausdorff.

A:

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: 25. For any set S CR, let S denote the union of all the open sets contained in S. (a) Prove that S…

A: The given question is solved below.

Q: Find an example of a nonempty set satisfying: 1. Closed, Dense in R 2. Open, Dense in R ~1. & 2.…

A:

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: 1. For each of the following sets, decide whether they are closed, bounded, compact (both) or…

A:

Q: Both questions below refer to set A = Z – {-2,0}. a) Give a partition of A into infinite many…

A: Solution: Definition: Let Ψ be a partion of the set A. That means, Ψ is the collection of subsets of…

Q: Determine (without proof) the suprema and infima of the following sets: (a) {n € N : n² < 10} (b)…

A:

Q: Suppose that you have a covering of a sphere by finitely many( closed) hemispheres. Prove that there…

A: Given that the sphere is covering by finitely many (closed) hemispheres.

Q: Given a finite set S, find all possible values of |S] so that the set {(a, b) | (a E S) A (b e S) ^…

A:

Q: Let A and B be any two subsets of a metric space (X, d). Then (1) A is a closed set. biso (2) If…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

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

A: Disconnected sets

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

A:

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: Let (S,≼) be a linearly ordered set with the Least-Upper-Bound Property. Let A and B be non-empty…

A: