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
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
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,
is dense-in-itself if every point of
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
.
Thus
is dense-in-itself if and only if
, where
is the
derived set of
.
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
contains at least one other irrational number
. 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
. As an example that is dense-in-itself but not dense in its topological space, consider