Verify that the two definitions of set 'A is nowhere dense' are equivalent: 1. A set is nowhere dense if it is not dense in any interval I, i.e. for any subinterval I of R, there is an (x,y) ⊂(I∩Ac ). 2. A set A is nowhere dense if the closure of A contains no nonempty open intervals.

Verify that the two definitions of set 'A is nowhere dense' are equivalent: 1. A set is nowhere dense if it is not dense in any interval I, i.e. for any subinterval I of R, there is an (x,y) ⊂(I∩Ac ). 2. A set A is nowhere dense if the closure of A contains no nonempty open intervals.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Let O be the collection of intervals Ia = (a, ∞) where a R along with I = 0 and I-∞ = R. Does this…

Q: 21. Let A and B be two on-empty bounded sets of positive embers and let С%3D {ху: х€ А and y € B}.…

Q: 1. For a subset YC (X,T ) , show that the collection TY ={Y NUJU ET} is closed under finite…

Q: Let X be an infinite set equipped with the indiscrete topology T, then (X,T) is compact. * True…

A: We have to determine the following statements are true or false : i) Let X be an infinite set…

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

Q: that the boundary of a closed set is nowhere dense in

A: Matric space

Q: Let A be a compact set in R. Show that the complement A is open and contains an interval of the form…

A: Solution: Before going into the problem we must know the Theoem named Heine-Borel Theorem.…

Q: let x be an infinite set and let ta be a topology on x in which all infinite subsets of x are open…

A: Given that X be an infinite set and let τ be a topology on X in which all infinite subsets of X are…

Q: Let A be any set with the discrete topology. Show that A is normal.

Q: A normed space X overR is separable if it has a conutable dense subset 0 صواب ihi

A: The given statement that " A normed space X over R is seperable if it has a countable dense subset "…

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

Q: ] Prove that a nonempty closed subset of R, if it is bounded from below, has a least element.

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

Q: prove that an is bounded above.

A: In this question, we prove the sequence an=(1+1n)n is convergent by given porcedure. i.e.…

Q: Use the Heine-Borel theorem to give an example of a set not compact in R with the usual topology…

A: An example of a set not compact in R with the usual topology and a non-compact set R2 with the usual…

Q: 3. Prove that if G is an open set dense in the metric space (S, d) then SG is nowhere dense in S.

Q: Prove that if A is a nonempty finite set and B is denumerable, then A × B is denumerable.

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

Q: 6 Prove that a closed set in the metric space (S, d) cither is nowhere dense in S or clse contains…

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

Q: Let X be an infinite set equipped with the indiscrete topology T, then (X,T) is compact. * True…

A: Note: We are entitled to solve only one question at a time. As the specified one is not mentioned,…

Q: Prove that for any infinite index set J, the uniform topology on R° is strictly finer than the…

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

A: The objective is to choose the correct option: Let X be a discrete space then a) X is homeomorphic…

Q: Prove the following sets are not compact by finding an open cover that does not have a finite…

A: a) Given- A set A=−1,1. Explanation- Consider a set Xn=-2,1-1n for n=1,2.... Then clearly,…

Q: prove that the outer measure of Cantor Set C (x), 0<x< 1, is 1-x the generalized

Q: Use the definition of compact set to prove that the set [0,1] is compact

A: We know that a set E is compact if every open cover of E has a finite subcover. We suppose, that O…

Q: Prove that the unoin of a finite number of bounded sets is bounded.

Q: Let T be a topology on X. Assume that T is Hausdorff and let x € X. (1) Show that {x} (and hence…

A: let T be a topology on x. T is a Hausdorff space

Q: Let S be a non-empty open subset of R that is bounded above. Show that sup S exists but that max S…

Q: Let A, BCR be two non-empty bounded sets. Assume that for every x E A, there exists y E B such that…

Q: Let X be a set with more than one element. (X,t) is a topological space such tha Vxe X,{x} €T then…

A: Given a set X with more than one element.

Q: Let S be a nonempty set of real numbers that is bounded from above and let x = sup S. Prove that…

Q: Suppose the collection = {B₁ (x): xe (0,1)}. Is B an open cover of the interval [0, 1) CR? Give an…

A: Given: The collection B=B12x : x∈[0, 1). To determine: Whether B is an open covering of the interval…

Q: Prove that if Gn is an open set which is dense in the complete metric space (S,d) for n = 1,2,...…

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: Suppose f(z) is holomorphic in an open connected set N. Can g(z) = f(z) be holomorphic in N? Explain…

Q: Assume X is Hausdorff. If K C X is compact and x ¢ K, show that there exist disjoint open sets U and…

Q: Let x be an infinite set equipped with the indiscrete topology T then (X,T) is compact. True or…

A: An infinite set equipped with discrete topology is not compact.

Q: Prove that a set S⊂R is everywhere dense iff it intersects every non-empty open sets in R. Prove…

Q: (b) Prove that the complement of a dense open subset of RR is nowhere dense in R.R..

A: (b) Prove that the complement of a dense open subset of R(R) is nowhere dense in R(R).

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: 4.Describe the largest set S on which it is correct to say that f is continuous S (x,y) = In(1–x² –…

A: Given problem:- Describe the largest set S on which it is correct to say that f is continuous C.…

Q: ndary of a path-connected set in R^2 with the usual

A: We have to check that boundary of a path connected set in R2 with the usual topology is…

Q: 28

A: Consider the interval [0,∞), the objective is to prove that the interval is not compact. Using the…

Q: A metric space is said to be separable if it contains a countable dense subset. (a) Is R separable?…

A: Set of all rational numbers Q belongs to R and Q is dense in R . Also set of all rational numbers Q…

Q: Let E be a set that bounded and c > 0. Prove that sup(cE) = -c inf(-E)

Q: 1. Let X C R"-1 be bounded. Prove that X considered as a subset of R" is a volume zero set.

A: Given, X⊆ℝn-1 That is, X is a subset of ℝ in terms of exponential n. So, let we consider the set,…

Question

Verify that the two definitions of set 'A is nowhere dense' are equivalent:

1. A set is nowhere dense if it is not dense in any interval I, i.e. for any subinterval I of R, there is an (x,y) ⊂(I∩A^c ).

2. A set A is nowhere dense if the closure of A contains no nonempty open intervals.

Expert Solution

Step 1

We will prove that (1) implies (2) and (2) implies (1). Thereby the two definitions will be equivalent.

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

For any infinite set X, the co-countable topology on X is defined to consist of all U in X so that either X\U is countable or U=0. Show that the co-countable topology satisfies the criteria for being a topology.
Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains some nonempty open set.
Define μ ∗ on the power set of R byμ∗(A) = (0 if A is bounded,1 if A is unbounded. Is μ∗ an outer measure or not? (Prove your claim.)
Use the Heine-Borel theorem to give an example of a set notcompact in R with the usual topology and a non-compact set R ^ 2 with the usual topology.
A metric space is said to be separable if it contains a countable dense subset. (a) Is R separable? (b) Is Rn separable?
Let X be an infinite set with the countable closed topology T={S subset of X :X_S is countable}. Then (X, T) is not connected?
Suppose A ⊂ R is both closed and bounded, prove that A has the Heine-Borel Property