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| In [[topology]] and related areas of [[mathematics]], a [[subset]] ''A'' of a [[topological space]] ''X'' is called '''dense''' (in ''X'') if every point ''x'' in ''X'' either belongs to ''A'' or is a [[limit point]] of ''A''.<ref name="CEIT">{{Citation |first=L. A. |last=Steen |first2=J. A. |last2=Seebach |title=[[Counterexamples in Topology]] |publisher=Dover |year=1995 |ISBN=0-486-68735-X}}</ref> Informally, for every point in ''X'', the point is either in ''A'' or arbitrarily "close" to a member of ''A'' - for instance, every [[real number]] is either a [[rational number]] or has one arbitrarily close to it (see [[Diophantine approximation]]).
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| Formally, a subset ''A'' of a topological space ''X'' is dense in ''X'' if for any point ''x'' in ''X'', any [[neighborhood (mathematics)|neighborhood]] of ''x'' contains at least one point from ''A''. Equivalently, ''A'' is dense in ''X'' if and only if the only [[closed set|closed subset]] of ''X'' containing ''A'' is ''X'' itself. This can also be expressed by saying that the [[closure (topology)|closure]] of ''A'' is ''X'', or that the [[interior (topology)|interior]] of the complement of ''A'' is empty.
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| The '''density''' of a topological space ''X'' is the least [[cardinality]] of a dense subset of ''X''.
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| ==Density in metric spaces==
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| An alternative definition of dense set in the case of [[metric space]]s is the following. When the topology of ''X'' is given by a metric, the [[topological closure|closure]] <math>\displaystyle \overline{A}</math> of ''A'' in ''X'' is the union of ''A'' and the set of all limits of sequences of elements in ''A'' (its ''limit points''),
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| :<math>\overline{A} = A \cup \{ \lim_n a_n : \forall n \ge 0, \ a_n \in A \}</math>
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| Then ''A'' is dense in ''X'' if
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| :<math>\overline{A} = X</math>
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| Note that <math>\scriptstyle A \;\subseteq\; \{ \lim_n a_n:\, \forall n \;\ge\; 0,\, \ a_n \in A \}</math>. If <math>\displaystyle \{U_n\}</math> is a sequence of dense [[open set|open]] sets in a complete metric space, ''X'', then <math>\displaystyle \cap^{\infty}_{n=1} U_n</math> is also dense in ''X''. This fact is one of the equivalent forms of the [[Baire category theorem]].
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| ==Examples==
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| The [[real number]]s with the usual topology have the [[rational number]]s as a [[countable set|countable]] dense subset which shows that the [[cardinality]] of a dense subset of a topological space may be strictly smaller than the cardinality of the space itself. The [[irrational number]]s are another dense subset which shows that a topological space may have several [[disjoint sets|disjoint]] dense subsets (in particular, two dense subsets may be each other's complements), and they need not even be of the same cardinality. Perhaps even more surprisingly, both the rationals and the irrationals have empty interiors, showing that dense sets need not contain any nonempty open set.
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| By the [[Weierstrass approximation theorem]], any given [[complex number|complex-valued]] [[continuous function]] defined on a [[closed interval]] [''a'',''b''] can be [[uniform convergence|uniformly approximated]] as closely as desired by a [[polynomial function]]. In other words, the polynomial functions are dense in the space C[''a'',''b''] of continuous complex-valued functions on the interval [''a'',''b''], equipped with the [[supremum norm]].
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| Every [[metric space]] is dense in its [[completion (metric space)|completion]].
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| == Properties ==
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| Every [[topological space]] is dense in itself. For a set ''X'' equipped with the [[discrete topology]] the whole space is the only dense set. Every non-empty subset of a set ''X'' equipped with the [[trivial topology]] is dense, and every topology for which every non-empty subset is dense must be trivial.
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| Denseness is [[transitive relation|transitive]]: Given three subsets ''A'', ''B'' and ''C'' of a topological space ''X'' with {{nowrap|''A'' ⊆ ''B'' ⊆ ''C''}} such that ''A'' is dense in ''B'' and ''B'' is dense in ''C'' (in the respective [[subspace topology]]) then ''A'' is also dense in ''C''.
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| The image of a dense subset under a [[surjective function|surjective]] continuous function is again dense. The density of a topological space (the least of the [[cardinality|cardinalities]] of its dense subsets) is a [[topological invariant]]. | |
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| A topological space with a [[connected space|connected]] dense subset is necessarily connected itself.
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| [[Continuous function]]s into [[Hausdorff space]]s are determined by their values on dense subsets: if two continuous functions {{nowrap|''f'', ''g'' : ''X'' → ''Y''}} into a [[Hausdorff space]] ''Y'' agree on a dense subset of ''X'' then they agree on all of ''X''.
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| == Related notions ==
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| A point ''x'' of a subset ''A'' of a topological space ''X'' is called a [[limit point]] of ''A'' (in ''X'') if every neighbourhood of ''x'' also contains a point of ''A'' other than ''x'' itself, and an [[isolated point]] of ''A'' otherwise. A subset without isolated points is said to be [[dense-in-itself]].
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| A subset ''A'' of a topological space ''X'' is called [[nowhere dense set|nowhere dense]] (in ''X'') if there is no neighborhood in ''X'' on which ''A'' is dense. Equivalently, a subset of a topological space is nowhere dense if and only if the interior of its closure is empty. The interior of the complement of a nowhere dense set is always dense. The complement of a closed nowhere dense set is a dense open set. Given a topological space ''X'', a subset ''A'' of ''X'' that can be expressed as the union of [[countable set|countably many]] nowhere dense subsets of ''X'' is called [[meagre set|meagre]]. The [[rational numbers]], while dense in the [[real numbers]], are meagre as a subset of the reals.
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| A topological space with a [[countable]] dense subset is called [[separable space|separable]]. A topological space is a [[Baire space]] if and only if the intersection of countably many dense open sets is always dense. A topological space is called [[resolvable space|resolvable]] if it is the union of two disjoint dense subsets. More generally, a topological space is called κ-resolvable if it contains κ pairwise disjoint dense sets.
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| An [[embedding]] of a topological space ''X'' as a dense subset of a [[compact space]] is called a [[compactification (mathematics)|compactification]] of ''X''.
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| A [[linear operator]] between [[topological vector space]]s ''X'' and ''Y'' is said to be [[densely defined operator|densely defined]] if its [[domain (mathematics)|domain]] is a dense subset of ''X'' and if its [[range (mathematics)|range]] is contained within ''Y''. See also [[continuous linear extension]].
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| A topological space ''X'' is [[hyperconnected space|hyperconnected]] if and only if every nonempty open set is dense in ''X''. A topological space is [[submaximal space|submaximal]] if and only if every dense subset is open.
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| ==See also==
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| *[[Dense order]]
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| ==References==
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| ===Notes===
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| {{Reflist}}
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| ===General references===
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| {{refbegin}}
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| * {{cite book |author=Nicolas Bourbaki |authorlink=Nicolas Bourbaki |title=General Topology, Chapters 1–4 |series=Elements of Mathematics |year=1989 |origyear=1971 |publisher=[[Springer-Verlag]] |isbn=3-540-64241-2 |ref=bourbaki}}
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| * {{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | origyear=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 | year=1995}}
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| {{refend}}
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| {{DEFAULTSORT:Dense Set}}
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| [[Category:General topology]]
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The main advantage of using the blog is that anyone can use the Word - Press blog and customize the elements in the theme regardless to limited knowledge about internet and website development. What I advise you do next is save the backup data file to a remote place like a CD-ROM, external disk drive if you have one or a provider such as Dropbox. The effect is to promote older posts by moving them back onto the front page and into the rss feed. s and intelligently including a substantial amount of key words in the title tags, image links, etc. Understanding how Word - Press works can be a challenge, but it is not too difficult when you learn more about it.
These folders as well as files have to copied and the saved. After all, Word - Press is free, many of the enhancements for Word - Press like themes and plugins are also free, and there is plenty of free information online about how to use Word - Press. We also help to integrate various plug-ins to expand the functionalities of the web application. This is identical to doing a research as in depth above, nevertheless you can see various statistical details like the number of downloads and when the template was not long ago updated. Once you've installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having to do all the coding yourself.
This gives a clearer picture that online shoppers are familiar with the WP ecommerce system. Word - Press has different exciting features including a plug-in architecture with a templating system. If Gandhi was empowered with a blogging system, every event in his life would have been minutely documented so that it could be recounted to the future generations. The animation can be quite subtle these as snow falling gently or some twinkling start in the track record which are essentially not distracting but as an alternative gives some viewing enjoyment for the visitor of the internet site. If you have any questions on starting a Word - Press food blog or any blog for that matter, please post them below and I will try to answer them.
Whether your Word - Press themes is premium or not, but nowadays every theme is designed with widget-ready. Russell HR Consulting provides expert knowledge in the practical application of employment law as well as providing employment law training and HR support services. Enterprise, when they plan to hire Word - Press developer resources still PHP, My - SQL and watch with great expertise in codebase. It supports backup scheduling and allows you to either download the backup file or email it to you. The Pakistani culture is in demand of a main surgical treatment.
Under Settings —> Reading, determine if posts or a static page will be your home page, and if your home page is a static page, what page will contain blog posts. By using Word - Press MLM websites or blogs, an online presence for you and your MLM company can be created swiftly and simply. In case you loved this post and you wish to receive more info with regards to backup plugin assure visit our webpage. It can be concluded that white label SEO comprise of a third party who resells a contract involving IT expert or consultant, SEO professional and end user. with posts or testimonials updated as they are uploaded to a particular section of the website. However, if you're just starting out your blog site or business site, you can still search for an ideal theme for it without breaking your bank account.