Sierpiński space

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Definition

The Sierpiński space is a topological space defined as follows (up to homeomorphism):

The underlying set is a two-point set $X=\{a,b\}$ .
The open subsets are: $\{\},\{a\},\{a,b\}$ . Thus, the closed subsets are $\{\},\{b\},\{a,b\}$ .

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.

Property	Satisfied?	Explanation	Corollary properties satisfied/dissatisfied
Separation
Kolmogorov space (the $T_{0}$ axiom)	Yes
T1 space	No	The subset $\{a\}$ is not closed.	dissatisfies: Hausdorff space
regular space	No	Consider the point $a$ and the closed subset $\{b\}$ . These cannot be separated by disjoint open subsets.
normal space	Yes	follows from being ultraconnected
Cardinality
finite space	Yes		satisfies: compact space and all corollaries thereof satisfies: second-countable space and all corollaries thereof
Connectedness
connected space	Yes
path-connected space	Yes	the function $f:[0,1]\to X$ that sends all elements to $a$ except 1 which is sent to $b$ is a continuous function.	satisfies: connected space
irreducible space	Yes	the only proper non-empty closed subset is $\{b\}$ , 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 $\{b\}$ , so the condition is vacuously satisfied.	satisfies: path-connected space, connected space, normal space
locally path-connected space	Yes		satisfies: locally connected space
Homotopy-invariant properties
contractible space	Yes		satisfies: weakly contractible space, simply connected space, acyclic space
Discreteness
discrete space	No
door space	Yes		satisfies: submaximal space, irresolvable space, hereditarily irresolvable space