Sierpiński space: Difference between revisions

Latest revision as of 18:04, 28 January 2012

This article is about a particular topological space (uniquely determined up to homeomorphism)|View a complete list of particular topological spaces

Definition

Explicit 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 ${}, {b}, {a, b}$ .

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
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

@@ Line 2: / Line 2: @@
 ==Definition==
+===Explicit definition===
 The '''Sierpiński space''' is a [[topological space]] defined as follows (up to [[homeomorphism]]):
@@ Line 7: / Line 9: @@
 * The underlying set is a two-point set <math>X = \{ a,b \}</math>.
 * The [[open subset]]s are: <math>\{ \}, \{ a \}, \{ a,b \}</math>. Thus, the closed subsets are <math>\{ \}, \{ b \}, \{ a,b \}</math>.
+===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==
@@ Line 18: / Line 24: @@
 |-
 | [[dissatisfies property::T1 space]] || No || The subset <math>\{ a \}</math> is not closed. || dissatisfies: [[dissatisfies property::Hausdorff space]]
+|-
+| [[dissatisfies property::regular space]] ||No || Consider the point <math>a</math> and the closed subset <math>\{ b \}</math>. These cannot be separated by disjoint open subsets. ||
+|-
+| [[satisfies property::normal space]] || Yes || follows from being ultraconnected ||
 |-
 ! Cardinality
@@ Line 27: / Line 37: @@
 | [[satisfies property::connected space]] || Yes || ||
 |-
-| [[satisfies property::path-connected space]] || Yes || the function <math>f:[0,1] \to X</math> that sends all elements to <math>a</math> except 1 which is sent to <math>b</math> is a continuous function. ||
+| [[satisfies property::path-connected space]] || Yes || the function <math>f:[0,1] \to X</math> that sends all elements to <math>a</math> except 1 which is sent to <math>b</math> is a continuous function. || satisfies: [[satisfies property::connected space]]
 |-
 | [[satisfies property::irreducible space]] || Yes || the only proper non-empty closed subset is <math>\{ b \}</math>, so the space cannot be expressed as a union of two such subsets. || satisfies: [[satisfies property::connected space]]
 |-
-| [[satisfies property::ultraconnected space]] || Yes || the only proper non-empty closed subset is <math>\{ b \}</math>, so the condition is vacuously satisfied. || satisfies: [[satisfies property::path-connected space]], [[satisfies property::connected space]]
+| [[satisfies property::ultraconnected space]] || Yes || the only proper non-empty closed subset is <math>\{ b \}</math>, so the condition is vacuously satisfied. || satisfies: [[satisfies property::path-connected space]], [[satisfies property::connected space]], [[satisfies property::normal space]]
+|-
+| [[satisfies property::locally path-connected space]] || Yes || || satisfies: [[satisfies property::locally connected space]]
+|-
+! Homotopy-invariant properties
+|-
+| [[satisfies property::contractible space]] || Yes || || satisfies: [[satisfies property::weakly contractible space]], [[satisfies property::simply connected space]], [[satisfies property::acyclic space]]
 |-
 ! Discreteness