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Concept
Sierpiński space
Summary
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.
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.
Explicitly, the Sierpiński space is a topological space S whose underlying point set is and whose open sets are
The closed sets are
So the singleton set is closed and the set is open ( is the empty set).
The closure operator on S is determined by
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
The Sierpiński space 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, has many properties in common with one or both of these families.
The points 0 and 1 are topologically distinguishable in S since 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
S is not regular (or completely regular) since the point 1 and the disjoint closed set 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 and cannot be precisely separated by a function. Indeed, cannot be the zero set of any continuous function since every such function is constant.
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.
Official source
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