This article is about a particular topological space (uniquely determined up to homeomorphism)|View a complete list of particular topological spaces
- The underlying set is a two-point set .
- The open subsets are: . Thus, the closed subsets are .
Definition as a left order topology
The Sierpiński space can be defined as the topological space arising by taking the left order topology on a totally ordered set of size two.
Topological space properties
|Property||Satisfied?||Explanation||Corollary properties satisfied/dissatisfied|
|Kolmogorov space (the axiom)||Yes|
|T1 space||No||The subset is not closed.||dissatisfies: Hausdorff space|
|regular space||No||Consider the point and the closed subset . These cannot be separated by disjoint open subsets.|
|normal space||Yes||follows from being ultraconnected|
|finite space||Yes|| satisfies: compact space and all corollaries thereof|
satisfies: second-countable space and all corollaries thereof
|path-connected space||Yes||the function that sends all elements to except 1 which is sent to is a continuous function.||satisfies: connected space|
|irreducible space||Yes||the only proper non-empty closed subset is , so the space cannot be expressed as a union of two such subsets.||satisfies: connected space|
|ultraconnected space||Yes||the only proper non-empty closed subset is , so the condition is vacuously satisfied.||satisfies: path-connected space, connected space, normal space|
|locally path-connected space||Yes||satisfies: locally connected space|
|contractible space||Yes||satisfies: weakly contractible space, simply connected space, acyclic space|
|door space||Yes||satisfies: submaximal space, irresolvable space, hereditarily irresolvable space|