1. J

    Why is the set of rational numbers dense, and set of integers numbers not?

    If we have two sets: Set one is the set of rational numbers with the usual less-than ordering Set two is the set of integers numbers with the usual less-than ordering Why is the set of rational numbers dense, and set of integers numbers not? Density is that for all choices of x and y with...
  2. I

    dense set

    Hello, I'm here to ask you your help. I have to do an exercise, but I have difficulties to finish. Let a and b be two reals strictly positive such that a/b is irrational. First, I have demonstrated that G=aZ+bZ is dense and then I have demonstrated that if A is a dense set and F a finish set...
  3. A

    Lebesque Measure

    Prove that there exists disjoint sets A and B for which [0, 1] = A U B where m(A) = 0 and B is the countable union of sets in [0, 1] which are nowhere dense in [0, 1].
  4. K

    Is there anything stopping a meager set from being complete?

    Is a complete meager set, or a complete nowhere dense set, a contradiction? Hello there, I just wrote a proof in which (X, ϱ) is a complete metric space which contain all bounded functions, f : N -> R. C is a subset of X, and C is defined such that it contain all bounded functions, f(n)...
  5. H

    Show simple functions are dense

    Let K be a compact space and let B be the space of bounded borel functions on K equipped with the supremum norm. Show that simple functions (i.e. functions attaining only a finite number of values) are dense in B. Maybe give a sketch proof and I fill in details Thanks
  6. L

    Dense sets

    Set I is nowhere dense if and only if A\I is dense for every dense set A. Prove.
  7. G

    An isometry onto a dense subset of equiv classes.

    I believe I have correctly solved most of the exercise in a text I am reading, but one portion is unclear to me. (X,d) is a metric space and S is the set of Cauchy sequences in X. (XBAR,rho) is the metric space of equivalence classes of the sequences in S, where rho(xbar,ybar) = lim...
  8. J

    Irrational flow yields dense orbits.

    I have the folloring problem: Given the following flow on the torus (Theta_1)' = w_1 and (Theta_2)' = w_2, where w_1 / w_2 is irrational then I am asked to show that each trajectory is DENSE. So I need to prove that Given any point p on the torus, any initial condition q, and any epsilon > 0...
  9. 4

    prove that D is dense

    Prove that a subset D of a metric space M is dense in M iff D \bigcap U nonempty for every nonempty open set U \subseteq M so given D \bigcap U nonempty we need to show that cl(D)=M, how does this work?
  10. H

    A Question About Nowhere Dense Sets

    I somehow come to the conclusion that the boundary of a set in a metric space is nowhere dense. Can anyone tell me if I'm right, and, if yes, kindly give me a hint on how to prove it! Thanks in advance!
  11. G

    Q is dense in R

    I have a problem here that has been giving me a little bit of trouble: Let (X,d) be a metric space. A subset S of X is said to be dense in X if S(bar) = X. Let S = Q and X = R. Prove that Q is dense in R. *S(bar) is the intersection of all the closed sets containing S. What is giving me a...
  12. C

    Dense set - problem

    Problem: Let p be an irrational number. Show that the set {exp(i*2PI*p*n), n natural number} is dense in the set {z, complex number: |z|=1}, (the unit circle). I am pretty lost, any hints or ideas? What I do know is the exp(i*2PI*p*n) is points on the unit circle, and then I´m stuck.
  13. H

    proof the set Q\N is dense in R

    please anyone can help me?? i dont even know how to start with... set of rationals not dense in
  14. B

    Separable polynomials are dense

    A separable polynomial is a polynomial which has no multiple roots in its splitting field. Let us consider polynomials of degree n as n-tuples over R (ie. as their coordinates relative to the basis {1,x,...,x^n}). We can define a neighbourhood of a polynomial using Euclidian distance : the...
  15. P

    Spec(R), dense points

    A point P in a topology for a set X is called dense if P is contained in every non-empty open set of the topology. Alternatively, the closure of \{P\} equals X. Find and prove a necessary and sufficient condition for \text{Spec}(R) to have a dense point. The condition should related to the...
  16. B

    Showing something is dense in ....

    x,y elements of R s.t x=y if |x-y|=k where k is an integer. => topological circle with circumference 1 Let B be an irrational real s.t. 0<B<1... look at the set B,2B,... and show that it is dense in 0<= x <= 1
  17. V

    Completely dense linear orderings

    Hello, There's a problem in some text I was reading about model theory, Show the theory of dense linear orderings without bounds is complete (and it gives a hint to use elimination of quantifiers). My understanding is that complete means every sentence in the logic can be proved or disproved...
  18. G

    f has non-isolated singularity. show that f is dense.

    More specifically, any help is greatly appreciated! thanks in advance.