Exercise 4. A subset A of a topological space X is called dense if cl A = X. The set Q of rational numbers and the set R\Q of irrational numbers are both subset of the real line. Show that they are both dense in the real line with its usual topology.

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: Does every bounded infinite set of real numbers have a limit point? Either prove that or give a…

A: Yes ; every bounded infinite set of real numbers has at least one limit point. Proof is given as…

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: 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 2.16. Suppose that A and B are countably infinite sets. (A) Show that AUB is countably…

A: Suppose A and B are two countably infinite sets.  To prove that A∪B is also countably infinite.…

Q: Q6) Find a topological space and a compact subset whose closure is not compact. Note hat a space is…

A:

Q: Prove

A:

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

A:

Q: Let ∅ be the empty set. Let X be the real line, the entire plane, or, in the three-dimensional case,…

A: Given:

Q: Let Y be an infinite subset of a compact set X ⊂ R. Prove Y' does not equal ∅.

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

A:

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: 4. Let A Sketch the collection of points Ar for i in the unit square.

A: Let A=1101. Sketch collection of points Ax¯ for x¯ in the unit square. Matrices maps a polygon to a…

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: 15. Let A be an infinite set. Prove that there exists an injective function f: N→ A.

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

A: Topology Question

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: Can a topology on an infinite set not be discrete and indiscrete in which every subset is clopen?

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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: Prove that the boundary points of an open interval (a, b) CR are the points a and b.

A: A point a∈ℝ is said to be boundary point of a set A⊂ℝ, if Ba,r∩A≠∅ and Ba,r∩Ac≠∅ for every…

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

A: Topology question.