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User:SamHB/Dense subset (view source)
Revision as of 03:41, December 15, 2015
, 03:41, December 15, 2015Restore a sandbox page that had been deleted by an admin in 2009.
{{Template:Math-e}}
If Q is a subset of R (think of R as the real numbers), we say that Q is '''dense''' in R if, wherever you are in R, there is a point in Q "infinitely close" to you. Put another way, you can never get any finite distance away from points in Q.
A classic example of a set that is '''not''' dense is the integers. If you are at 1.414, there is no integer "infinitely close" to you— the nearest one is .414 away from you.
A classic example (in fact, '''the''' classic example) of a set that '''is''' dense is the set of [[rational numbers]], that is, fractions of integers. There is a rational that exactly matches 1.414: <math>\frac{1414}{1000}</math>. But how about a number that is not itself rational, like <math>\sqrt{2}</math>? This is where the subtle aspects of "infinitely close" come into play. What this means is that, for any closeness that you want (but not exact equality), there is a rational number that close. If you want a rational number within one thousandth, <math>\frac{1414}{1000}</math> will do. If you want a rational number within one trillionth, <math>\frac{1414213562373}{1000000000000}</math> will do.
==In multiple dimensions==
Subsets can be dense in two (or more) dimensions. In the plane (the set of points given by two real numbers—the x-coordinate and the y-coordinate), the points with both coordinates rational are dense. If you are at <math>(\sqrt{2}, \sqrt{3})</math>, the rational point <math>(\frac{14142}{10000}, \frac{17321}{10000})</math> is within one-thousandth of where you are.
==More precise definition==
{{Template:Math-h}}
The precise definition of what we meant above by "as close as you want", uses the "<math>\varepsilon\,</math>" concept used in [[Limit (mathematics)|limits]], which is a concept of [[Calculus]]. Q is dense in R if, for any point x in R and any <math>\varepsilon > 0\,</math>, there is a point in Q that is within <math>\varepsilon\,</math> of x.
==Metric spaces==
In more advanced calculus treatments (this subject is often called [[Analysis]]), denseness can be defined in any [[Metric space]]. The notion of a point being "within <math>\varepsilon\,</math> of x" simply uses the metric.
==Countability==
{{Template:Math-a}}
The fact that the rationals are both countable and dense is very important in mathematical analysis. There exist sets (not the reals, of course) that have no countable dense sets.
==Topology==
In the language of topology, many of the above concepts are recast in different terms. Q is dense in R if the [[closure]] of Q is R.
[[category:mathematics]]
If Q is a subset of R (think of R as the real numbers), we say that Q is '''dense''' in R if, wherever you are in R, there is a point in Q "infinitely close" to you. Put another way, you can never get any finite distance away from points in Q.
A classic example of a set that is '''not''' dense is the integers. If you are at 1.414, there is no integer "infinitely close" to you— the nearest one is .414 away from you.
A classic example (in fact, '''the''' classic example) of a set that '''is''' dense is the set of [[rational numbers]], that is, fractions of integers. There is a rational that exactly matches 1.414: <math>\frac{1414}{1000}</math>. But how about a number that is not itself rational, like <math>\sqrt{2}</math>? This is where the subtle aspects of "infinitely close" come into play. What this means is that, for any closeness that you want (but not exact equality), there is a rational number that close. If you want a rational number within one thousandth, <math>\frac{1414}{1000}</math> will do. If you want a rational number within one trillionth, <math>\frac{1414213562373}{1000000000000}</math> will do.
==In multiple dimensions==
Subsets can be dense in two (or more) dimensions. In the plane (the set of points given by two real numbers—the x-coordinate and the y-coordinate), the points with both coordinates rational are dense. If you are at <math>(\sqrt{2}, \sqrt{3})</math>, the rational point <math>(\frac{14142}{10000}, \frac{17321}{10000})</math> is within one-thousandth of where you are.
==More precise definition==
{{Template:Math-h}}
The precise definition of what we meant above by "as close as you want", uses the "<math>\varepsilon\,</math>" concept used in [[Limit (mathematics)|limits]], which is a concept of [[Calculus]]. Q is dense in R if, for any point x in R and any <math>\varepsilon > 0\,</math>, there is a point in Q that is within <math>\varepsilon\,</math> of x.
==Metric spaces==
In more advanced calculus treatments (this subject is often called [[Analysis]]), denseness can be defined in any [[Metric space]]. The notion of a point being "within <math>\varepsilon\,</math> of x" simply uses the metric.
==Countability==
{{Template:Math-a}}
The fact that the rationals are both countable and dense is very important in mathematical analysis. There exist sets (not the reals, of course) that have no countable dense sets.
==Topology==
In the language of topology, many of the above concepts are recast in different terms. Q is dense in R if the [[closure]] of Q is R.
[[category:mathematics]]