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dense set∗ yark† 2013-03-21 13:02:42 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 ε > 0 and all x ∈ X there is y ∈ D such that d(x, y) < ε. For example, both the rationals Q and the irrationals R \ Q are dense in the reals 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 ℵ0 if it would otherwise be finite; with this convention, the spaces of density ℵ0 are precisely the separable spaces. The density of a topological space X is denoted d(X). If d(X) X is a Hausdorff space, it can be shown that |X| ≤ 22 . ∗ hDenseSeti created: h2013-03-21i by: hyarki version: h31192i Privacy setting: h1i hDefinitioni h54A99i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1