# Dense subset

In the mathematical subject of topology , a dense subset of a metric or topological space is a subset of this space with special properties. The term dense subset is defined in its general form in topology. It is also used in many other sub-disciplines of mathematics, such as analysis , functional analysis and numerics , for example in the approximation of continuous functions using polynomials .

A subset is said to be dense in a metric space if one can approximate every point of the total space as precisely as required by a point from the subset. The rational numbers thus form a dense subset of the set of real numbers . This means that irrational numbers can be approximated with arbitrary precision using rational fractions or finite decimal numbers . More generally, it is said of a subset , they lie closely in a topological space , where each environment of an arbitrary point from always one element from contains. ${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle \ mathbb {R}}$${\ displaystyle A}$${\ displaystyle X}$${\ displaystyle x}$${\ displaystyle X}$${\ displaystyle A}$

## Definition in metric spaces

Given a metric space (such as a normalized space with the metric ). ${\ displaystyle (X, d)}$ ${\ displaystyle (X, \ | \ cdot \ |)}$${\ displaystyle d (x, y) = \ | xy \ |}$

Then a set is called dense in if one of the following equivalent statements is true: ${\ displaystyle M \ subseteq X}$${\ displaystyle X}$

• For each and every one there is a point , so that is.${\ displaystyle x \ in X}$${\ displaystyle r> 0}$${\ displaystyle y \ in M}$${\ displaystyle d (x, y)
• For each and every one there is a point , so that is. In this case, referred to the open ball order of radius .${\ displaystyle x \ in X}$${\ displaystyle r> 0}$${\ displaystyle y \ in M}$${\ displaystyle y \ in B_ {r} (x)}$${\ displaystyle B_ {r} (x)}$${\ displaystyle x}$${\ displaystyle r}$
• For each there is a sequence of points such that is.${\ displaystyle x \ in X}$${\ displaystyle (x_ {n}) _ {n \ in \ mathbb {N}}}$${\ displaystyle M}$${\ displaystyle \ textstyle \ lim _ {n \ to \ infty} x_ {n} = x}$
• The closed shell of the crowd is the whole space, so .${\ displaystyle M}$${\ displaystyle {\ overline {M}} = X}$

The above definition by the limit value of a sequence cannot be transferred to general topological spaces. The convergence of sequences has to be generalized for this by the filter convergence or the convergence of networks .

## Examples

• The set of rational numbers is close to the set of real numbers .${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle \ mathbb {R}}$
• The set of irrational numbers is close to the set of real numbers .${\ displaystyle \ mathbb {R}}$
• The set of polynomials is close to the set of continuous functions on a compact interval.
• The set of test functions is close to the set of Lebesgue integrable functions.
• Let be a subset of a space normalized by means . If one denotes the closed envelope of this set with respect to the norm , then lies in .${\ displaystyle M}$${\ displaystyle \ | \ cdot \ |}$ ${\ displaystyle X}$${\ displaystyle {\ overline {M}}}$ ${\ displaystyle \ | \ cdot \ |}$${\ displaystyle M}$${\ displaystyle {\ overline {M}}}$
• The set of natural numbers is not close to the set of rational numbers , it is even nowhere close to .${\ displaystyle \ mathbb {N}}$${\ displaystyle \ mathbb {Q}}$${\ displaystyle \ mathbb {Q}}$
• The Cantor set is an uncountable, closed and nowhere dense subset of the real numbers.
• The interval is not dense in the real numbers, but neither is it dense anywhere, because it is dense in what is a neighborhood of zero.${\ displaystyle [-1.1]}$${\ displaystyle [-1.1]}$
• The space of the smooth functions with a compact carrier lies close to the space of the square-integrable functions .${\ displaystyle C_ {c} ^ {\ infty} (\ mathbb {R} ^ {n})}$${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle L ^ {2} (\ mathbb {R} ^ {n})}$

## Definition in topological spaces

A topological space is given . Then a set is dense if and only if one of the following equivalent conditions is met: ${\ displaystyle (X, {\ mathcal {O}})}$${\ displaystyle M}$${\ displaystyle X}$

• The ending of corresponds to the superset, so it applies .${\ displaystyle M}$${\ displaystyle {\ overline {M}} = X}$
• The crowd cuts any non-empty open set, so it's for everyone .${\ displaystyle M}$${\ displaystyle M \ cap O \ neq \ emptyset}$${\ displaystyle O \ in {\ mathcal {O}}}$
• Each environment in contains a point from .${\ displaystyle X}$${\ displaystyle M}$

A set is called dense in if it is dense with respect to the subspace topology . Sometimes the dense sets in the superset are also called dense everywhere . ${\ displaystyle M}$${\ displaystyle Y \ subset X}$ ${\ displaystyle {\ mathcal {O}} _ {Y}}$${\ displaystyle X}$

## properties

• Inclusion: if it is tight in and , so is also in .${\ displaystyle A}$${\ displaystyle X}$${\ displaystyle A \ subseteq B \ subseteq X}$${\ displaystyle B}$${\ displaystyle X}$
• Transitivity: if it is dense in and dense in , it is already dense in .${\ displaystyle X}$${\ displaystyle Y}$${\ displaystyle Y}$${\ displaystyle Z}$${\ displaystyle X}$${\ displaystyle Z}$
• Conservation under continuous mapping: if dense is in and a continuous mapping is dense in .${\ displaystyle A}$${\ displaystyle X}$${\ displaystyle f \ colon X \ to Y}$${\ displaystyle f (A)}$${\ displaystyle f (X)}$

The last property is provided with the subspace topology of ; the concept of the dense subset is then to be understood with reference to this subspace topology. ${\ displaystyle f (X)}$${\ displaystyle Y}$

## Linearly ordered sets

A special case of the topological term dense results from its application to ordered sets. A subset of a strictly totally ordered set is dense (in ) if it at all and made with one out there, so . This special case arises from the order topology on and will be explained in more detail there. ${\ displaystyle S}$${\ displaystyle (M, <)}$${\ displaystyle M}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle M}$${\ displaystyle x ${\ displaystyle z}$${\ displaystyle S}$${\ displaystyle x ${\ displaystyle M}$

## Partially ordered sets

A different topology is common in partially ordered sets used in forcing theory. For a partially ordered set , the sets (for ) form the basis of a topology . A set closely related if and only if for every element of there is an element which satisfies. ${\ displaystyle (P, \ leq)}$${\ displaystyle U_ {p}: = \ {q \ in P \ mid q \ leq p \}}$${\ displaystyle p \ in P}$${\ displaystyle \ tau _ {\ leq}}$${\ displaystyle D \ subseteq P}$${\ displaystyle \ tau _ {\ leq}}$${\ displaystyle p}$${\ displaystyle P}$${\ displaystyle d \ in D}$${\ displaystyle d \ leq p}$

### Nowhere dense crowds

A nowhere dense set is a subset of a topological space in which the interior of its closure is empty. So it applies ${\ displaystyle A}$

${\ displaystyle \ operatorname {Int} ({\ overline {A}}) = \ emptyset}$.

Contrary to their name, nowhere are dense sets not the opposite or complement of dense or everywhere dense sets. More precisely, a set is nowhere dense if and only if it is not dense in any (non-empty) open set. Thus, dense sets are never dense anywhere, since they are always dense in the open set . Conversely, however, there are both non-dense sets that are not dense anywhere (like the integers in ) and non-dense sets that are not dense anywhere (like the interval in .) ${\ displaystyle X}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle [2,3]}$${\ displaystyle \ mathbb {R}}$

### Separable and Polish rooms

A topological space is called a separable space if it contains a countable, dense set. This often facilitates the demonstration, so separable rooms are "easier" to handle. The concept of Polish space is even stronger ; this is a topological space that contains a countable, dense subset and is completely metrizable .