Exercise 6: [Challenge] Suppose that you have a covering of a sphere by finitely many (closed) hemispheres. Prove that there is a subcollection of less than five hemispheres that cover the sphere. Remark: A set is called closed if it contains its boundary points. In the case of a hemisphere, this means it contains the boundary (great) circle. Compare to a closed interval, e.g., the interval 0, 1] is closed since it contains the boundary points of 0 and 1. Last modified: October 2, 2019, Due: October 9, 2019. 1 Hint 1: Consider the case of five hemispheres. Prove that four hemispheres suffice. Hint 2: Consider the case of a circle: Show that if four semicircles cover a circle, then three of them suffice.

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: 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: are closed.

A:

Q: Every point in space is colored red, green, or blue. Prove or disprove that there exists a…

A: Given that every point in space is colored red, green or blue.  Using pigeon hole principle let's…

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: Consider the following statements: (i) If A is not compact then there are infinitely many open…

A: Consider the following statements: (i) If A is not compact then there are infinitely many open…

Q: Prove the following: The countable product of metrizable spaces is a metrizable space.

A:

Q: Exercise 4. A subset A of a topological space X is called dense if cl A = X. The set Q of rational…

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: The set of all points in a plane equidistant from a given point is a ( Select ) . The given point is…

A: The solution is:

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

A:

Q: frnd an o-thogonal besis for the {(0,2,1),,0,1), (lo,4.0), (4,2,2){ space spanned by dac set

A:

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

A: Topology Question

Q: Suppose that B is infinite and that A is a finite subset of B. Then B − A is infinite

A: Let A and B be two sets such that A is a finite subset of B and B is infinite. Prove B-A is…

Q: Show that an open subcover R is not compact by finding cover that does not have finite

A:

Q: thun no in

A:

Q: every finite subset of metric space is compact. ture or false?

A:

Q: Show that both ℝ and ℝ2 may be covered by countably many open spheres.

A: We have to show that  and  covered by countably many open balls.

Q: Every point in the plane is colored using one of 2020 colors. Prove or disprove that there exists a…

A: Using the pigeon hole principle let's prove that can find a rectangle with vertices of same color.…

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: ": Prove that ( R", ||||) is complete metric space.

A:

Q: Determine whether each product is an inner product space in R3. If not, explain why. 1) = u1v1 +…

A: