Theorem 2.1. Let{U}{_, be a finite collection of open sets in a topological space (X,T). Then N, U is open. Why does your proof not prove the false statement that the infinite intersection of open sets is necessarily open? n

Q: Theorem 3.7: A subset A ofa metric space (X, d) is closed if and only if /.contains all its limit…

A: We have to prove that A is Closed if and only if A contains all its limit points. Note : In proof i…

Q: Ametric space (x,d) is complete if and only ifx has the cantor intersection property

A:

Q: 3.1 Show that an infinite intersection of closed sets Fg, k = 1, 2, 3, ..., in a metric space (S, d)…

A: to prove 3.1 first we will prove that infinite union of open sets in a metric space ( S, d ) is open…

Q: 3.1 Show that an infinite intersection of closed sets E k = 1,2,3,., in a metric space (SA) is a…

A:

Q: 2. (a) Use axiom (2) in the definition of a topology to show that if {U1,U2, . ,Un} is a finite…

A: It is given in part (a) that (X,t) be any topological space, and the interval (x-A,x+A) is only open…

Q: Theorem 3.2.14. (i) The union of a finite collection of closed sets is closed.

A: we have to show that ; The union of a finite collection of closed set is closed.

Q: Let E1, E2, ..., En ⊂ X,d be compact subsets of a metric space X. Just using the definition of…

A: For a metric space we know that the for a compact set every open cover has a finite subcover. So…

Q: 3.2 Prove that in any metric space (S.) every closed ball Se[xo] is a closed set.

A:

Q: Theorem 4.4.8. Let (S, d) be a metric space. A set A is open if and only if its complement R\A is…

A: Given that a<b in ℝ The objective is to determine whether the given sets are open or not.

Q: Theorem 2.26. Let A be a subset of a topological space X. Then p is an interior point of A if and…

A: First well show that, if there existsan open set U with P∈UCA, thenP will be an interior point of…

Q: Theorem 6.20 (Heine-Borel Theorem). Let A be a subset of R" with the standard topol- ogy. Then A is…

A: Given :- Let A be a subset of ℝn with the standard topology. To Proof :- A is compact if and only if…

Q: Let X be a topological space. Prove that (a) Ø and X are closed. (b) Intersections of any number of…

A: Union of any open sets is open and intersection of two open sets is open.

Q: Theorem 3.7: A subset A of a metric space (X, d) is closed if and only if/.contains all its limit…

A: A subset A of a metric space (X, d) is closed iff A contains all its limit points.

Q: Theorem 8.6. Let C be a connected subset of the topological space X. If D is a subset of X such that…

A: Let C be a connected subset of the topological space X.Also given that D is a subset of X such that…

Q: Theorem 2.28. Let A be a subset of a topological space X. Then Int(A), Bd(A), and Int(X – A) are…

A: Given that A is a subset of the topological space X. We have to prove that IntA, BdA and IntX-A are…

Q: 2.8.12 Let X be a metric space. Prove that an arbitrary intersection of compact subsets of X is…

A:

Q: let C be a compact subset Tn a Haus dorft topolopfcal space (K,R) and bex a If bac then prove that…

A:

Q: Theorem 3.16. Suppose X is a set and S is a collection of subsets of X. Then S is a subbasis for…

A: Given that X is a set and S is a collection of subsets of X. Let B be the family of finite…

Q: Theorem 6.3. If X is a compact space, then every infinite subset of X has a limit point.

A: We have given that, X is Compact space.

Q: Theorem 3.5. Finite Intersection Property If {Ka} is a collection of compact subsets of a metric…

A:

Q: Prove that every finite T,- space has the discrete topology.

A: T=(X, τ)  (X, τ) is a T1 space if and only if all points of X are closed in T.   Let X≠∅ be a set.…

Q: Prove or disprove that (R, Tco-finite) is T2- space

A: It is given that ℝ,τco-finite. τco-finite=U⊂ℝ   such that  X-U is either finite or X  To check…

Q: 4. Let (X, d) be a metric space and xnx where each x, e X and x e X. Let A be the subset of X which…

A:

Q: Theorem 2.25. Suppose that (X, p) is a metric space. (a) The intersection of any collection of…

A:

Q: a) Consider the set X = {1,2,3} with the topology T = {Ø, X,{1}, {2, 3}, {1,2, 3}}. Show that (X, t)…

A: As per our guidelines we are supposed to answer only one asked question.Kindly repost other…

Q: Show that any infinite subset A of a discrete topological space X is not compact.

A:

Q: Theorem 2.3. A set U is open in a topological space (X,J) if and only if for every point x€ U, there…

A:

Q: Let X and Y be two discrete spaces, then * X is never homeomorphic to Y X is homomorphic to Y if and…

A:

Q: in the discrete metric space: 1. All collections are clo

A: Let (X, d) be a metric space. Consider a closed ball Brx0 with both center x0 ∈ X and radius r >…

Q: Let X and Y be two discrete spaces, then * X is homomorphic to Y if and only if X and Y are both…

A: Our guidelines we are supposed to answer only one question. Kindly repost other question as the next…

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

A:

Q: 1. Let M, d be a metric space with A C B C M. Suppose that B is totally bounded. Prove that A is…

A: Let (M, d) be a metric space and let A be a nonempty subset of M. The diameter of A is defined as…

Q: Prove that the open set (0,2) is not a compact subset of ℝ by finding an open cover with no finite…

A: To prove:  (0, 2) is not a compact set of ℝ.

Q: Let M, be set of all rare subsets of a metric space X. Prove that countable union of M, is rare.

A: As per bartleby guidelines for more than one question asked only first should be answered. Please…

Q: A topological space (X, t) is a T1-space iff t contains the cofinite topology on X.

A:

Q: Theorem 2.15. Let (X,T) be a topological space, and let U be an open set and A a closed subset of X.…

A:

Q: Let X be a discrete spaces then * X is homeomorphic to R if and only if X is finite X is…

A: Given that, X is a discrete space. To be homeomorphic X has to be of same cardinality as ℝ and so X…

Q: Theorem 6.17. Every compact subset C of R contains a maximum in the set C, i.e., there is an m E C…

A: To prove that, every compact subset C of ℝ contains a maximum in the set C, that is, there is an m∈C…

Q: Every closed subset of a connected metric space is connected. O True O False

A:

Q: a) Consider the set X = {1,2,3} with the topology t= (0,X, (1). (2, 3}. (1, 2, 3} }. Show that (X,r)…

A:

Q: (a) Show that every metric space is T2-space.

A:

Q: let T be the finite closed topology on any set X.Every subset of (X,T) is connected true or false?

A: let T be the finite closed topology on any set X.Every subset of (X,T) is connected true or false?

Q: Theorem 8.11. For topological spaces X and Y, X ×Y is connected if and only if each of X and Y is…

A:

Q: Theorem 6.5. A space X is compact if and only if every collection of closed sets with the finite…

A:

Q: Let E,F ⊆ X,d be compact subsets of a metric space X. Just using the definition of compactness…

A:

Q: 1/ For any. topological space (X, T), if T has only three elements then (X, T) is connected.

A: Given: Topological space X, T and T has 3 elements  To prove that X, T is connected when T has 3…

Q: Let X be a discrete spaces then * X is never homeomorphic to R O X is homeomorphic to R if and only…

A: Fourth option is correct.

Q: Let E1, E2 be two compact subsets of RP. Use the Heine-Borel Theorem to prove that EU É, is also a…

A:

Percentage

A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.

Algebraic Expressions

In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.

Numbers

Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.

Subtraction

Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.

Addition

Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.