Show that any convex open set is simply connected.

Q: {1, x + x², x²} ≤ P2. Show that this is a spanning set for P2

A:

Q: If finite set A has a elements and finite set B has b elements, and a> b what is the minimum number…

A: Given that set A has a elements,  n(A)=a and set B has b elements,  n(B)=b with a>b .We have to…

Q: Does there exist a non-empty open set A and two elements in A that are farthest apart (among all…

A: There are no two elements that are farthest apart in a non-empty open set. Justification : Let A…

Q: Suppose a set M is closed and bounded. Prove that M has a maximal and minimal element.

A: Suppose a set M is closed and bounded. Then we have to show that M has a maximal and minimal…

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: Provide a counter example that infinity intersection of open set is open

A:

Q: Show that compact sets are closed under arbitrary intersections and finite unions. (Here we're…

A: To show that compact sets are closed under arbitrary intersections and finite unions

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: Are   any   open   subsets   of   R   sequentially   compact?     Explain   why   or   why not

A: The objective of the question is to determine whether any open subsets of the set of real numbers…

Q: A closed set in the finite complement topology on R.

A: Def: A set C is closed in topological space, if complement of C is open set.

Q: Use the definition of compactness to prove that a closed subset Y of a compact set X ⊆ R is compact

A: In this question, we use the definition of compactness to prove that a closed subset Y of a compact…

Q: If A C BỊ U B2 where B1 and B2 are disjoint open sets and A is compact, show that An B1 is compact.…

A: To Show- If A ⊆ B1 ∪ B2 where B1 and B2 are disjoint open sets and A is compact, show that A ∩ B1 is…

Q: prove that Cantor set is closed and compact.

A: Use the property that arbitrary intersection of closed set is closed and the Heine Borel Theorem.

Q: 4. Prove that the intersection of an arbitrary collection of compact sets in R is compact.

A: (Heine-Borel Theorem). A subset A of R is compact if and only if A is bounded and closed (with…

Q: Call a set A CR clopen if it is both open and closed. Prove that the only clopen sets are Ø and R.

A: A⊆R clopen if it is both open and closed.  We have to prove that the only clopen sets are ϕ and R

Q: Let A and B be open sets in a topological space. Is A ∩ B open?

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: Prove that a set is open iff it contains none of its boundary points; i.e., A is openAnA = 0.

A:

Q: Prove or disprove whether or not the following two-dimensional set is convex.  S = { (x1,x2 ) such…

A:

Q: Prove that the convex hull of a bounded set is bounded.

A:

Q: Given a compact set A and a closed set B. ShOw that AB is compact!

A: A is a compact set B is closed

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

A: