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dense set (Definition)

A subset $ D$ of a topological space $ X$ is said to be dense (or everywhere dense) in $ X$ if the closure of $ D$ is equal to $ X$. Equivalently, $ D$ is dense if and only if $ D$ intersects every nonempty open set.

In the special case that $ X$ is a metric space with metric $ d$, then this can be rephrased as: for all $ \varepsilon > 0$ and all $ x\in X$ there is $ y\in D$ such that $ d(x,y)<\varepsilon$.

For example, both the rationals $ \mathbb{Q}$ and the irrationals $ \mathbb{R} \setminus \mathbb{Q}$ are dense in the reals $ \mathbb{R}$.

The least cardinality of a dense set of a topological space is called the density of the space. It is conventional to take the density to be $ \aleph_0$ if it would otherwise be finite; with this convention, the spaces of density $ \aleph_0$ are precisely the separable spaces. The density of a topological space $ X$ is denoted $ d(X)$. If $ X$ is a Hausdorff space, it can be shown that $ \vert X\vert \le 2^{2^{d(X)}}$.



"dense set" is owned by yark. [ full author list (2) | owner history (1) ]
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See Also: nowhere dense, dense (in a poset)

Other names:  dense subset, everywhere dense set, everywhere dense subset, everywhere-dense set, everywhere-dense subset
Also defines:  dense, everywhere dense, everywhere-dense, density

Attachments:
Hewitt-Marczewski-Pondiczery theorem (Theorem) by yark
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Cross-references: Hausdorff space, separable spaces, finite, cardinality, reals, dense in, irrationals, rationals, metric, metric space, open set, intersects, closure, topological space, subset
There are 45 references to this entry.

This is version 8 of dense set, born on 2002-01-03, modified 2007-06-19.
Object id is 1192, canonical name is Dense.
Accessed 13816 times total.

Classification:
AMS MSC: 54A99 (General topology :: Generalities :: Miscellaneous)
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