Concept# Dense set

Summary

In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).
Formally, A is dense in X if the smallest closed subset of X containing A is X itself.
The of a topological space X is the least cardinality of a dense subset of X.
Definition
A subset A of a topological space X is said to be a of X if any of the following equivalent conditions are satisfied:
The smallest closed subset of X containing A is X itself.
The closure of A in X

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Gabriel Nivasch, János Pach, Gábor Tardos

Given a collection of n opaque unit disks in the plane, we want to find a stacking order for them that maximizes their visible perimeter, the total length of all pieces of their boundaries visible from above. We prove that if the centers of the disks form a dense point set, i.e., the ratio of their maximum to their minimum distance is O(n^1/2), then there is a stacking order for which the visible perimeter is Ω(n^2/3). We also show that this bound cannot be improved in the case of the n^1/2×n^1/2 piece of a sufficiently small square grid. On the other hand, if the set of centers is dense and the maximum distance between them is small, then the visible perimeter is O(n^3/4) with respect to any stacking order. This latter bound cannot be improved either. These results partially answer some questions of Cabello, Haverkort, van Kreveld, and Speckmann.

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Given a collection of n opaque unit disks in the plane, we want to find a stacking order for them that maximizes their visible perimeter, the total length of all pieces of their boundaries visible from above. We prove that if the centers of the disks form a dense point set, i.e., the ratio of their maximum to their minimum distance is O(n(1/2)), then there is a stacking order for which the visible perimeter is Omega(n(2/3)). We also show that this bound cannot be improved in the case of a sufficiently small n(1/2) x n(1/2) uniform grid. On the other hand, if the set of centers is dense and the maximum distance between them is small, then the visible perimeter is O(n(3/4)) with respect to any stacking order. This latter bound cannot be improved either. Finally, we address the case where no more than c disks can have a point in common. These results partially answer some questions of Cabello, Haverkort, van Kreveld, and Speckmann. (C) 2013 Elsevier B.V. All rights reserved.

Maria Colombo, Luigi De Rosa, Massimo Sorella

In this work we show that, in the class of L-infinity((0,T); L-2(T-3)) distributional solutions of the incompressible Navier-Stokes system, the ones which are smooth in some open interval of times are meagre in the sense of Baire category, and the Leray ones are a nowhere dense set.