Dense-in-itself

 ( Topology ) 

In general topology, a subset $A$ of a topological space is said to be dense-in-itselfSteen & Seebach, p. 6Engelking, p. 25 or crowded if $A$ has no isolated point. Equivalently, $A$ is dense-in-itself if every point of $A$ is a limit point of $A$. Thus $A$ is dense-in-itself if and only if $A\backslash subseteq\; A\text{'}$, where $A\text{'}$ is the derived set of $A$.

A dense-in-itself closed set is called a perfect set. (In other words, a perfect set is a closed set without isolated point.)

The notion of dense set is distinct from dense-in-itself. This can sometimes be confusing, as " X is dense in X" (always true) is not the same as " X is dense-in-itself" (no isolated point).

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Examples A simple example of a set that is dense-in-itself but not closed (and hence not a perfect set) is the set of irrational numbers (considered as a subset of the real numbers). This set is dense-in-itself because every neighborhood of an irrational number $x$ contains at least one other irrational number $y\; \backslash neq\; x$. On the other hand, the set of irrationals is not closed because every rational number lies in its closure. Similarly, the set of rational numbers is also dense-in-itself but not closed in the space of real numbers.

The above examples, the irrationals and the rationals, are also in their topological space, namely $\backslash mathbb\{R\}$. As an example that is dense-in-itself but not dense in its topological space, consider $\backslash mathbb\{Q\}\; \backslash cap\; 0,1$. This set is not dense in $\backslash mathbb\{R\}$ but is dense-in-itself.

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Properties A singleton subset of a space $X$ can never be dense-in-itself, because its unique point is isolated in it.

The dense-in-itself subsets of any space are closed under unions.Engelking, 1.7.10, p. 59 In a dense-in-itself space, they include all .Kuratowski, p. 78 In a dense-in-itself T₁ space they include all .Kuratowski, p. 78 However, spaces that are not T₁ may have dense subsets that are not dense-in-itself: for example in the dense-in-itself space $X=\backslash \{a,b\backslash \}$ with the indiscrete topology, the set $A=\backslash \{a\backslash \}$ is dense, but is not dense-in-itself.

The closure of any dense-in-itself set is a perfect set.Kuratowski, p. 77

In general, the intersection of two dense-in-itself sets is not dense-in-itself. But the intersection of a dense-in-itself set and an open set is dense-in-itself.

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See also

• Nowhere dense set
• Glossary of topology
• Dense order

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Notes

• Ryszard Engelking (1989). 9783885380061, Heldermann Verlag, Berlin. ISBN 9783885380061
• Kuratowski, K. (1966). 012429202X, Academic Press. 012429202X
• (1978). 9780486687353, Springer-Verlag. ISBN 9780486687353

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