• Source: Dense-in-itself

In general topology, a subset



A


{\displaystyle A}

of a topological space is said to be dense-in-itself or crowded
if



A


{\displaystyle A}

has no isolated point.
Equivalently,



A


{\displaystyle A}

is dense-in-itself if every point of



A


{\displaystyle A}

is a limit point of



A


{\displaystyle A}

.
Thus



A


{\displaystyle A}

is dense-in-itself if and only if



A
⊆

A
′



{\displaystyle A\subseteq A'}

, where




A
′



{\displaystyle A'}

is the derived set of



A


{\displaystyle 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).




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


{\displaystyle x}

contains at least one other irrational number



y
≠
x


{\displaystyle y\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 dense sets in their topological space, namely




R



{\displaystyle \mathbb {R} }

. As an example that is dense-in-itself but not dense in its topological space, consider




Q

∩
[
0
,
1
]


{\displaystyle \mathbb {Q} \cap [0,1]}

. This set is not dense in




R



{\displaystyle \mathbb {R} }

but is dense-in-itself.




Properties


A singleton subset of a space



X


{\displaystyle 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. In a dense-in-itself space, they include all open sets. In a dense-in-itself T1 space they include all dense sets. 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
=
{
a
,
b
}


{\displaystyle X=\{a,b\}}

with the indiscrete topology, the set



A
=
{
a
}


{\displaystyle A=\{a\}}

is dense, but is not dense-in-itself.
The closure of any dense-in-itself set is a perfect set.
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.




See also


Nowhere dense set
Glossary of topology
Dense order




Notes






References


Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Berlin. ISBN 3-88538-006-4.
Kuratowski, K. (1966). Topology Vol. I. Academic Press. ISBN 012429202X.
Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1978). Counterexamples in Topology (Dover reprint of 1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 0507446.
This article incorporates material from Dense in-itself on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Kata Kunci Pencarian:

• Astrofisika
• Dense-in-itself
• Dense set
• Dense order
• Density (disambiguation)
• Isolated point
• Derived set (mathematics)
• Glossary of general topology
• Nowhere dense set
• Perfect set
• Vertical draft