Dense-in-itself
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In
general topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
, a subset A of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
is said to be dense-in-itself or crowded if A has no
isolated point ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equival ...
. Equivalently, A is dense-in-itself if every point of A is a
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
of A. Thus A is dense-in-itself if and only if A\subseteq A', where A' is the derived set of A. A dense-in-itself
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
is called a
perfect set In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set S is perfect if S=S', where S' denotes the set of all Limit point, limit points of S, also known as the derived se ...
. (In other words, a perfect set is a closed set without isolated point.) The notion of
dense set In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the r ...
is unrelated to ''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 In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
(considered as a subset of the
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
). This set is dense-in-itself because every
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of an irrational number x contains at least one other irrational number y \neq x. On the other hand, the set of irrationals is not closed because every
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
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 set In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the r ...
s in their topological space, namely \mathbb. As an example that is dense-in-itself but not dense in its topological space, consider \mathbb \cap ,1/math>. This set is not dense in \mathbb but is dense-in-itself.


Properties

A
singleton Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics), a set with exactly one element * Singleton field, used in conformal field theory Computing * Singleton pattern, a design pattern that allows only one instance ...
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. In a dense-in-itself space, they include all
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
s. In a dense-in-itself T1 space they include all
dense set In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the r ...
s. However, spaces that are not T1 may have dense subsets that are not dense-in-itself: for example in the space X=\ with the
indiscrete topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
, the set A=\ is dense, but is not dense-in-itself. The closure of any dense-it-itself set is a
perfect set In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set S is perfect if S=S', where S' denotes the set of all Limit point, limit points of S, also known as the derived se ...
.Kuratowski, p. 77


See also

*
Nowhere dense set In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. ...
*
Glossary of topology This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fund ...
*
Dense order In mathematics, a partial order or total order < on a X is said to be dense if, for all x


Notes


References

* * * {{PlanetMath attribution, id=6228, title=Dense in-itself Topology