# Nowhere dense sub set of unit circle

This is identical to showing that the fractional parts of n 2 π are dense in [ 0, 1), which is true since 1 2 π is irrational (see e.g. this question). In mathematics, a nowhere dense set on a topological space is a set whose closure has empty . For example, if X is the unit interval [0,1], not only is it possible to have a dense set of Lebesgue measure zero (such as the set of rationals), but it. ¯A = X. A set A ⊂ X is called nowhere dense if X \ ¯A is everywhere dense. . between the interval with identified endpoints (Example ) and the unit circle.

Then (i) either S = S", that is, the set S coincides with the invariant unit circle S1, ( ii) or the set S is a Cantor set in S", that is, S is nowhere dense on the invariant. Then {A | supp(d,u>\,pp) O Iint I 0} is a countable union of nowhere dense subsets of 6D. Remark. By the discussion above, that means {A | supp(du>(,pp) O Iint I. Let I be a closed interval in 83 with nonempty interior. Let I C supp(efyi). Then {A | supp(cfyi.\,pp) fl 7lnt ^ 0} is a countable union of nowhere dense subsets of

A countable union of nowhere dense sets can still be dense. surface, R. Suppose that the cluster set of f at every point on the unit circle is R. Then there is . Terminology: Y is a nowhere dense subset of X, when intY = ∅. It is meagre B( X), C(K), L∞, c0, c, ℓ∞, H∞ (bounded analytic functions on the unit disc D with. where X is either the closed unit interval or the circle, then the irregular set of h is nowhere dense in X. It can be easily shown that, for compact spaces, the. A nowhere dense set A, has the property that for each open subset of . with a irrational, defines a transitive rotation of the unit circle in the complex plane.