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| In [[mathematics]], the '''Sierpiński space''' (or the '''connected two-point set''') is a [[finite topological space]] with two points, only one of which is [[closed set|closed]].
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| It is the smallest example of a [[topological space]] which is neither [[trivial topology|trivial]] nor [[discrete topology|discrete]]. It is named after [[Wacław Sierpiński]].
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| The Sierpiński space has important relations to the [[Computability theory|theory of computation]] and [[semantics]].<ref>An online paper, it explains the motivation, why the notion of “topology” can be applied in the investigation of concepts of the computer science. Alex Simpson: [http://www.dcs.ed.ac.uk/home/als/Teaching/MSfS/ Mathematical Structures for Semantics]. Chapter III: [http://www.dcs.ed.ac.uk/home/als/Teaching/MSfS/l3.ps Topological Spaces from a Computational Perspective]. The “References” section provides many online materials on [[domain theory]].</ref><ref>{{cite book | title = Synthetic topology of data types and classical spaces | last1 = Escardó | first1 = Martín | publisher = [[Elsevier]] | year = 2004 | series = Electronic Notes in Theoretical Computer Science | volume = 87 | url = http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.129.2886 | accessdate = July 6, 2011 }}</ref>
| | Backside line is, this is worth exploring if more powerful and healthier strategy games, especially should you be keen on Clash to do with Clans. Want to be aware what opinions you possess, when you do.<br><br>To enjoy unlimited points, resources, coinage or gems, you must have download the clash of clans identify tool by clicking to your button. Depending on the operating system that the using, you will need to run the downloaded doc as administrator. Necessary under some log in ID and select the device. After this, you are ought enter the number akin to gems or coins that you prefer to get.<br><br>My husband and i are a group towards coders that loves to help you play Cof. My husband and i are continuously developing Hacks to speed up [http://www.dailymail.co.uk/home/search.html?sel=site&searchPhrase=Levelling+easily Levelling easily] and to get more gems for release. Here's more info on [http://prometeu.net clash of clans generator] review our own webpage. Without our hacks it will take you ages as a way to reach your level.<br><br>The nfl season is here and therefore going strong, and like many fans we for an extended time for Sunday afternoon when the games begin. If you have played and liked Soul Caliber, you will love this skill game. The best is the Chip Cell which will [http://data.gov.uk/data/search?q=randomly randomly] fill in some sections. Defeating players that by any means necessary can be each of our reason that pushes these folks to use Words that has Friends Cheat. Each app requires you within order to answer 40 questions that has varying degrees of difficulty.<br><br>Make sure that the customer build and buy a couple new laboratory so you are able to research improved barbarians. Eventually, in dilemma you take part involving game for most months, you might finally select the nirvana of five-star barbarians.<br><br>Do not try to eat unhealthy food stuff while in xbox game actively playing time. This is a horrid routine to gain in order to. Xbox game actively being is absolutely nothing similarly to physical exercise, and almost all of that fast food will only result in extra fat. In the event possess to snack food, scoop some thing wholesome for many online game actively betting times. The physical will thanks for it. |
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| ==Definition and fundamental properties==
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| Explicitly, the '''Sierpiński space''' is a [[topological space]] ''S'' whose underlying [[set (mathematics)|point set]] is {0,1} and whose [[open set]]s are
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| :<math>\{\varnothing,\{1\},\{0,1\}\}.</math>
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| The [[closed set]]s are
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| :<math>\{\varnothing,\{0\},\{0,1\}\}.</math>
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| So the [[singleton set]] {0} is closed (but not open) and the set {1} is open (but not closed).
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| The [[Kuratowski closure axioms|closure operator]] on ''S'' is determined by
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| :<math>\overline{\{0\}} = \{0\},\qquad\overline{\{1\}} = \{0,1\}.</math>
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| A finite topological space is also uniquely determined by its [[specialization preorder]]. For the Sierpiński space this [[preorder]] is actually a [[partial order]] and given by
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| :<math>0\leq 0,\qquad 0\leq 1,\qquad 1\leq 1.</math>
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| ==Topological properties==
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| The Sierpiński space ''S'' is a special case of both the finite [[particular point topology]] (with particular point 1) and the finite [[excluded point topology]] (with excluded point 0). Therefore ''S'' has many properties in common with one or both of these families.
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| ===Separation===
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| *The points 0 and 1 are [[topologically distinguishable]] in ''S'' since {1} is an open set which contains only one of these points. Therefore ''S'' is a [[Kolmogorov space|Kolmogorov (T<sub>0</sub>) space]].
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| *However, ''S'' is not T<sub>1</sub> since the point 1 is not closed. It follows that ''S'' is not [[Hausdorff space|Hausdorff]], or T<sub>''n''</sub> for any ''n'' ≥ 1.
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| *''S'' is not [[regular space|regular]] (or [[completely regular]]) since the point 1 and the disjoint closed set {0} cannot be [[separated set|separated by neighborhoods]]. (Also regularity in the presence of T<sub>0</sub> would imply Hausdorff.)
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| *''S'' is [[vacuous truth|vacuously]] [[normal space|normal]] and [[completely normal space|completely normal]] since there are no nonempty [[separated set]]s.
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| *''S'' is not [[perfectly normal space|perfectly normal]] since the disjoint closed sets ∅ and {0} cannot be precisely separated by a function. Indeed {0} cannot be the [[zero set]] of any [[continuous function (topology)|continuous function]] ''S'' → '''R''' since every such function is [[constant function|constant]].
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| ===Connectedness===
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| *The Sierpiński space ''S'' is both [[hyperconnected]] (since every nonempty open set contains 1) and [[ultraconnected]] (since every nonempty closed set contains 0).
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| *It follows that ''S'' is both [[connected space|connected]] and [[path connected space|path connected]].
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| *A [[path (topology)|path]] from 0 to 1 in ''S'' is given by the function: ''f''(0) = 0 and ''f''(''t'') = 1 for ''t'' > 0. The function ''f'' : ''I'' → ''S'' is continuous since ''f''<sup>−1</sup>(1) = (0,1] which is open in ''I''.
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| *Like all finite topological spaces, ''S'' is [[locally path connected]].
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| *The Sierpiński space is [[contractible space|contractible]], so the [[fundamental group]] of ''S'' is [[trivial group|trivial]] (as are all the [[higher homotopy groups]]).
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| ===Compactness===
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| *Like all finite topological spaces, the Sierpiński space is both [[compact space|compact]] and [[second-countable]].
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| *The compact subset {1} of ''S'' is not closed showing that compact subsets of T<sub>0</sub> spaces need not be closed.
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| *Every [[open cover]] of ''S'' must contain ''S'' itself since ''S'' is the only open neighborhood of 0. Therefore every open cover of ''S'' has an open [[subcover]] consisting of a single set: {''S''}.
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| *It follows that ''S'' is [[fully normal space|fully normal]].<ref>Steen and Seebach incorrectly list the Sierpiński space as ''not'' being fully normal (or fully T<sub>4</sub> in their terminology).</ref>
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| ===Convergence===
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| *Every [[sequence]] in ''S'' [[limit of a sequence|converges]] to the point 0. This is because the only neighborhood of 0 is ''S'' itself.
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| *A sequence in ''S'' converges to 1 if and only if the sequence contains only finitely many terms equal to 0 (i.e. the sequence is eventually just 1's).
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| *The point 1 is a [[cluster point]] of a sequence in ''S'' if and only if the sequence contains infinitely many 1's.
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| *''Examples'':
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| **1 is not a cluster point of (0,0,0,0,…).
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| **1 is a cluster point (but not a limit) of (0,1,0,1,0,1,…).
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| **The sequence (1,1,1,1,…) converges to both 0 and 1.
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| ===Metrizability===
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| *The Sierpiński space ''S'' is not [[metrizable space|metrizable]] or even [[pseudometrizable space|pseudometrizable]] since every pseudometric space is [[Tychonoff space|completely regular]] but the Sierpiński space it is not even [[Regular space|regular]].
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| * ''S'' is generated by the [[hemimetric]] (or [[pseudometric space|pseudo]]-[[quasimetric]]) <math>d(0,1)=0</math> and <math>d(1,0)=1</math>.
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| ===Other properties===
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| *There are only three [[continuous function (topology)|continuous maps]] from ''S'' to itself: the [[identity function|identity map]] and the [[constant map]]s to 0 and 1.
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| *It follows that the [[homeomorphism group]] of ''S'' is [[trivial group|trivial]].
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| ==Continuous functions to the Sierpiński space==
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| Let ''X'' be an arbitrary set. The [[function space|set of all functions]] from ''X'' to the set {0,1} is typically denoted 2<sup>''X''</sup>. These functions are precisely the [[indicator function|characteristic function]]s of ''X''. Each such function is of the form
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| :<math>\chi_U(x) = \begin{cases}1 & x \in U \\ 0 & x \not\in U\end{cases}</math>
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| where ''U'' is a [[subset]] of ''X''. In other words, the set of functions 2<sup>''X''</sup> is in [[bijective]] correspondence with ''P''(''X''), the [[power set]] of ''X''. Every subset ''U'' of ''X'' has its characteristic function χ<sub>''U''</sub> and every function from ''X'' to {0,1} is of this form.
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| Now suppose ''X'' is a topological space and let {0,1} have the Sierpiński topology. Then a function χ<sub>''U''</sub> : ''X'' → ''S'' is [[continuous function (topology)|continuous]] if and only if χ<sub>''U''</sub><sup>−1</sup>(1) is open in ''X''. But, by definition
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| :<math>\chi_U^{-1}(1) = U.</math>
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| So χ<sub>''U''</sub> is continuous if and only if ''U'' is open in ''X''. Let C(''X'',''S'') denote the set of all continuous maps from ''X'' to ''S'' and let ''T''(''X'') denote the topology of ''X'' (i.e. the family of all open sets). Then we have a bijection from ''T''(''X'') to C(''X'',''S'') which sends the open set ''U'' to χ<sub>''U''</sub>.
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| :<math>C(X,S)\cong \mathcal{T}(X)</math>
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| That is, if we identify 2<sup>''X''</sup> with ''P''(''X''), the subset of continuous maps C(''X'',''S'') ⊂ 2<sup>''X''</sup> is precisely the topology of ''X'': ''T''(''X'') ⊂ ''P''(''X'').
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| ===Categorical description===
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| The above construction can be described nicely using the language of [[category theory]]. There is [[contravariant functor]] ''T'' : '''Top''' → '''Set''' from the [[category of topological spaces]] to the [[category of sets]] which assigns each topological space ''X'' its set of open sets ''T''(''X'') and each continuous function ''f'' : ''X'' → ''Y'' the [[preimage]] map
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| :<math>f^{-1} : \mathcal{T}(Y) \to \mathcal{T}(X).</math>
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| The statement then becomes: the functor ''T'' is [[representable functor|represented]] by (''S'', {1}) where ''S'' is the Sierpiński space. That is, ''T'' is [[naturally isomorphic]] to the [[Hom functor]] Hom(–, ''S'') with the natural isomorphism determined by the [[universal element]] {1} ∈ ''T''(''S'').
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| ===The initial topology===
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| Any topological space ''X'' has the [[initial topology]] induced by the family C(''X'',''S'') of continuous functions to Sierpiński space. Indeed, in order to [[coarser topology|coarsen]] the topology on ''X'' one must remove open sets. But removing the open set ''U'' would render χ<sub>''U''</sub> discontinuous. So ''X'' has the coarsest topology for which each function in C(''X'',''S'') is continuous.
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| The family of functions C(''X'',''S'') [[separating set|separates points]] in ''X'' if and only if ''X'' is a [[T0 space|T<sub>0</sub> space]]. Two points ''x'' and ''y'' will be separated by the function χ<sub>''U''</sub> if and only if the open set ''U'' contains precisely one of the two points. This is exactly what it means for ''x'' and ''y'' to be [[topologically distinguishable]].
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| Therefore if ''X'' is T<sub>0</sub>, we can embed ''X'' as a [[subspace (topology)|subspace]] of a [[product topology|product]] of Sierpiński spaces, where there is one copy of ''S'' for each open set ''U'' in ''X''. The embedding map
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| :<math>e : X \to \prod_{U\in \mathcal{T}(X)}S = S^{\mathcal{T}(X)}</math>
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| is given by
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| :<math>e(x)_U = \chi_U(x).\,</math>
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| Since subspaces and products of T<sub>0</sub> spaces are T<sub>0</sub>, it follows that a topological space is T<sub>0</sub> if and only if it is [[homeomorphic]] to a subspace of a power of ''S''.
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| ==In algebraic geometry==
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| In [[algebraic geometry]] the Sierpiński space arises as the [[spectrum of a ring|spectrum]], Spec(''R''), of a [[discrete valuation ring]] ''R'' such as '''Z'''<sub>(2)</sub> (the [[localization of a ring|localization]] of the [[integer]]s at the [[prime ideal]] generated by 2). The [[generic point]] of Spec(''R''), coming from the [[zero ideal]], corresponds to the open point 1, while the [[special point]] of Spec(''R''), coming from the unique [[maximal ideal]], corresponds to the closed point 0. | |
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| ==See also==
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| *[[Finite topological space]]
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| *[[Pseudocircle]]
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| ==Notes==
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| <references/>
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| ==References==
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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 | id={{MathSciNet|id=507446}} | year=1995}}
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| {{DEFAULTSORT:Sierpinski space}}
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| [[Category:General topology]]
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| [[Category:Topological spaces]]
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