In mathematics, the Sierpiński space is a finite topological space with two points, only one of which is closed. It is the smallest example of a topological space which is neither trivial nor discrete. It is named after Wacław Sierpiński. The Sierpiński space has important relations to the theory of computation and semantics, because it is the classifying space for open sets in the Scott topology.
Definition and fundamental properties Explicitly, the Sierpiński space is a topological space S whose underlying point set is { 0 , 1 } {\displaystyle \{0,1\}} and whose open sets are
{ ∅ , { 1 } , { 0 , 1 } } . {\displaystyle \{\varnothing ,\{1\},\{0,1\}\}.}
The closed sets are
{ ∅ , { 0 } , { 0 , 1 } } . {\displaystyle \{\varnothing ,\{0\},\{0,1\}\}.}
So the singleton set { 0 } {\displaystyle \{0\}} is closed and the set { 1 } {\displaystyle \{1\}} is open ( ∅ = { } {\displaystyle \varnothing =\{\,\}} is the empty set). The closure operator on S is determined by
{ 0 } ¯ = { 0 } , { 1 } ¯ = { 0 , 1 } . {\displaystyle {\overline {\{0\}}}=\{0\},\qquad {\overline {\{1\}}}=\{0,1\}.}
A finite topological space is also uniquely determined by its specialization preorder. For the Sierpiński space this preorder is actually a total order and given by
0 ≤ 0 , 0 ≤ 1 , 1 ≤ 1. {\displaystyle 0\leq 0,\qquad 0\leq 1,\qquad 1\leq 1.}
Topological properties The Sierpiński space S {\displaystyle 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 {\displaystyle S} has many properties in common with one or both of these families.
Separation The points 0 and 1 are topologically distinguishable in S since { 1 } {\displaystyle \{1\}} is an open set which contains only one of these points. Therefore, S is a Kolmogorov (T0) space. However, S is not T1 since the point 1 is not closed. It follows that S is not Hausdorff, or Tn for any n ≥ 1. {\displaystyle n\geq 1.}
S is not regular (or completely regular) since the point 1 and the disjoint closed set { 0 } {\displaystyle \{0\}} cannot be separated by neighborhoods. (Also regularity in the presence of T0 would imply Hausdorff.) S is vacuously normal and completely normal since there are no nonempty separated sets. S is not perfectly normal since the disjoint closed sets ∅ {\displaystyle \varnothing } and { 0 } {\displaystyle \{0\}} cannot be precisely separated by a function. Indeed, { 0 } {\displaystyle \{0\}} cannot be the zero set of any continuous function S → R {\displaystyle S\to \mathbb {R} } since every such function is constant.
Connectedness The Sierpiński space S is both hyperconnected (since every nonempty open set contains 1) and ultraconnected (since every nonempty closed set contains 0). It follows that S is both connected and path connected. A path from 0 to 1 in S is given by the function: f ( 0 ) = 0 {\displaystyle f(0)=0} and f ( t ) = 1 {\displaystyle f(t)=1} for t > 0. {\displaystyle t>0.} The function f : I → S {\displaystyle f:I\to S} is continuous since f − 1 ( 1 ) = ( 0 , 1 ] {\displaystyle f^{-1}(1)=(0,1]} which is open in I. Like all finite topological spaces, S is locally path connected. The Sierpiński space is contractible, so the fundamental group of S is trivial (as are all the higher homotopy groups).
Compactness Like all finite topological spaces, the Sierpiński space is both compact and second-countable. The compact subset { 1 } {\displaystyle \{1\}} of S is not closed showing that compact subsets of T0 spaces need not be closed. 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 } . {\displaystyle \{S\}.}
It follows that S is fully normal.
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