Exercise 2.5. Verify that the discrete, indiscrete, finite complement, and countable com- plement topologies are indeed topologies for any set X.

Q: Prove that a subset of a set of measure zero has measure zero.

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Q: Show that S CR is a closed set if and only if S contains all of its limit points.

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Q: 4. (a) If X is an infinite dimensional space, show that the set {x = x | || x || = 1} is not…

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Q: Exercise 2.15. Construct uncountable sets A and B such that A\ B is: (A) finite (B) countably…

A: See the attachment

Q: Let S be a nonempty subset in R". Prove or disprove by a counterexample that Span S = (S+)+

A: We have to prove or disprove by counterexample that Span S=S⊥⊥

Q: Give an example of a set in R ^ 2 with the usual topology that is not connected but has a connected…

A: Let's examine the  problem for path-connectedness example of a set in R2 with the usual topology…

Q: Let Y be an infinite subset of a compact set X c R. Prove Y' # Ø.

A: Y' denotes the derived set of Y i.e. the set of all limit points of Y. We just need to show that any…

Q: Show for every nonempty, finite set E that supE = maxE.

A: Given, E is a finite set. E is a non-empty set. To prove, supE = maxE We know that, sup is supremum…

Q: Let x E A. Prove that A is an infinite set if and only if A≈ (A - {x}).

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Q: Suppose that A and B are bounded, nonempty subsets of R. Prove that AU B is bounded and that sup(AU…

A: Consider the provided question, Suppose A and B are bounded, nonempty subsets of R.

Q: Consider the following topologies on R: • Ti = the trivial topology • T2 = the standard topology •…

A: We know that trivial topology on X is τ1={∅,X}. It is the most scratchy of them all. For the…

Q: Prove a neighborhood is an open set

A: Recall that, a neighbourhood of a point is a set of points containing that point where one can move…

Q: Exercise 2.46. Show that: (1) The topology generated by a subbasis is a topology. (2) The sets in…

A: 2.46

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: Prove or disprove. A closed and bounded subset of e' is compact.

A: PROVE OR DISPROVE :-  A closed and bounded subset of l1 is compact.

Q: (b) Show that the continuous image of compact set is compact but the convers need not to be true.…

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Q: the collection of all close subsets of R is called the topology of R

A: To check: Whether the statement is true or false: Given: The statement is "The collection of all…

Q: (1) Let X be a set and tEX, let t={GCX/ t€G or G° finite }, show that whether t is a topology on X…

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Q: Prove that any closed interval [s, t] can be written as the intersection of a collectable collection…

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Q: Let X be an infinite set. (a) Show that is a topology on X, called the finite co со

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Q: If B1,..., Bn is a finite open cover of a compact set A, can the union B1 U ·. U Bn equal A exactly?…

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Q: 2. Is there a set A that satisfies A = {A}? If yes, exhibit one %3D such. If not, Why not exactly?

A: No, A≠{A} for any set A. Let  A=1,2,3 then number of elements in A are 3. Now,  A=1,2,3 here the…

Q: Let X be an infinite set T be topology of the a Linfinite subsets that is on x- and let on X in…

A: Given : X be an infinite set and T be a topology on X such that all infinite subsets of X are open.…

Q: Prove that T is the discrete topology for X iff every point in X is an open set

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Q: For any infinite set X, the co-countable topology on X is defined to consist of all U in X so that…

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Q: Give an example of a closed set in R ^ 3 with the usual topology that is not compact

A: Given the usual topology on ℝ3. i.e. τusual=U⊆ℝ3: ∀x∈U, ∃∈>0 such that B∈x⊆U

Q: If B1,..., Bn is a finite open cover of a compact set A, can the union B1 U... U Bn equal A exactly?…

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Q: 3.6 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

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Q: If M and N are two norms on space X then prove that M and N are equivalent if and only if a subset…

A: Any two norms M and N on a norm space is called equivalent if  M≤N≤M. in the given question we have…

Q: If X #0, compare the following topologies on X: a) The co-finite topology 7, and the co-countable…

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Q: 4. Use the definition of compactness to prove that a closed subset Y of a compact set X CR is…

A: We will answer the first question since we answer only one question at a time. Please resubmit the…

Q: Prove that the Cantor set has measure zero.

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Q: Use only the definition of affine dependence to show that an indexed set {v1, v} in R" is affinely…

A:

Q: if

A: R is not connected if T is the finite closed topology.

Q: In R with the cofinite topology, every preopen set is open

A: As we know that , In ℝ with discrete topology of every semi open set is open.

Q: Let A be a nonempty set and define B={kx : x∈A}, where k≥0 is fixed. Prove that supB=ksupA.

A: Given A be a nonempty set and define B=kx:x∈A where k≥0 is fixed. We have to prove that supB=ksupA…

Q: Let X be an infinite set with the finite closed topology T={S subset of X; X-S is finite}. Then

A: a) If not, then there are non-empty disjoint open sets  such that  . Since they are disjoint, so  .…

Q: Let ACR be a nonempty bounded set. Prove that sup A is unique.

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Q: Show that the derived set E′ of any set E is closed.

A: Here is the proof .

Q: df. Every nonempty proper subset of R is either open or closed.

A: False

Q: Let Z be with the co-finite topology If A = {1,3,5, 7,.. }, then A A O O O

A: As Z is with co-finite topology. Every open neighborhood of an infinite set must contain infinitely…

Q: show that empty set, X is an element of S intersection T in topology

A: Here We have to show that empty set must be in S∩T where S and T are topology on set X.

Q: 4. If X #0, compare the following topologies on X: a) The co-finite topology Tf and the co-countable…

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Q: show by an example that a nonempty proper subset of S in the metric space (S, d) can be both open…

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Q: Show that X is connected if and only if the only clopen subsets are Ø and X.

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Q: Exercise 60 Let p and q be two points in a topological space X. Show that there exists an open set U…

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Q: Drag the correct description of the transformation and congruence criterion into the empty box below…

A: Given,

Q: B. Prove that the union of a finite number of compact subsets F1, F2, ..., F, of RP is a compact…

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Q: Show that every closed subset F of R' is the intersection of a countable col- lection of open sets.

A: detailed explanation mentioned below.

Q: Consider the set X={a,b,c,d} construct all the possible topologies of X that have only three open…

A: Given: The set X=a,b,c,d To find:  all the possible topologies of X that has only three open…

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.