Dense

Every subset of a nowhere dense set is nowhere dense, and the union of finitely many nowhere dense sets is nowhere dense.

Ambergris is less dense than water and floats, whereas amber is too dense to float, though less dense than stone.

Open and closed * A nowhere dense set need not be closed (for instance, the set is nowhere dense in the reals), but is properly contained in a nowhere dense closed set, namely its closure (which would add 0 to the set).

The surrounding space matters: a set A may be nowhere dense when considered as a subspace of a topological space X but not when considered as a subspace of another topological space Y. A nowhere dense set is always dense in itself.

Dense to very dense fog is also expected in the morning.

On Sunday, it warned of dense to very dense fog conditions in the region for the next 48 hours.

This image provided by the UW/Fred Hutch Cancer Center Breast Imaging in October 2024 shows mammogram scans with dense and not dense tissue.

As per the weather report, dense to very dense fog may be expected in some parts of North India.

It is even more dense than gold and about twice as dense as silver.

Articular ligaments "Ligament" most commonly refers to a band of dense regular connective tissue bundles made of collagenous fibers, with bundles protected by dense irregular connective tissue sheaths.

Cold, dense water is shown in blue, flowing south from upper latitudes, while warm, less dense water, shown in red flows north from low latitudes. citation The upper layers are formed by deep open ocean convection during winter.

Compared to other dense subsets of the real line, such as the rational numbers, the dyadic rationals are in some sense a relatively "small" dense set, which is why they sometimes occur in proofs.

Hence, the graph is dense and we can use any algorithm which works on Dense graphs in time O(m).

Indeed, a set is nowhere dense if and only if its closure is nowhere dense.

It can be seen in a pot of boiling water in which the hot and less-dense water on the bottom layer moves upwards in plumes, and the cool and more dense water near the top of the pot likewise sinks.

More generally, if is any infinite cardinal, then a product of at most spaces with dense subsets of size at most has itself a dense subset of size at most (Hewitt–Marczewski–Pondiczery theorem).

On the other hand, water and most oils are immiscible, usually forming layers with the least dense liquid as the top layer, and the most dense layer at the bottom.

The irrational numbers have the same property, but the Cantor set has the additional property of being closed, so it is not even dense in any interval, unlike the irrational numbers which are dense in every interval.

The set i(X) is then a dense subset of Y. If X is Hausdorff, then i is an isomorphism onto i(X), and thus X can be identified with a dense subset of its completion.

The union of countably many nowhere dense sets, however, need not be nowhere dense.