Alexandroff plank
Alexandroff plank in topology, an area of mathematics, is a topological space that serves as an instructive example.
Definition
The construction of the Alexandroff plank starts by defining the topological space [math]\displaystyle{ (X,\tau) }[/math] to be the Cartesian product of [math]\displaystyle{ [0,\omega_1] }[/math] and [math]\displaystyle{ [-1,1], }[/math] where [math]\displaystyle{ \omega_1 }[/math] is the first uncountable ordinal, and both carry the interval topology. The topology [math]\displaystyle{ \tau }[/math] is extended to a topology [math]\displaystyle{ \sigma }[/math] by adding the sets of the form [math]\displaystyle{ U(\alpha,n) = \{p\} \cup (\alpha,\omega_1] \times (0,1/n) }[/math] where [math]\displaystyle{ p = (\omega_1,0) \in X. }[/math]
The Alexandroff plank is the topological space [math]\displaystyle{ (X,\sigma). }[/math]
It is called plank for being constructed from a subspace of the product of two spaces.
Properties
The space [math]\displaystyle{ (X,\sigma) }[/math] has the following properties:
- It is Urysohn, since [math]\displaystyle{ (X,\tau) }[/math] is regular. The space [math]\displaystyle{ (X,\sigma) }[/math] is not regular, since [math]\displaystyle{ C = \{(\alpha,0) : \alpha \lt \omega_1\} }[/math] is a closed set not containing [math]\displaystyle{ (\omega_1,0), }[/math] while every neighbourhood of [math]\displaystyle{ C }[/math] intersects every neighbourhood of [math]\displaystyle{ (\omega_1,0). }[/math]
- It is semiregular, since each basis rectangle in the topology [math]\displaystyle{ \tau }[/math] is a regular open set and so are the sets [math]\displaystyle{ U(\alpha,n) }[/math] defined above with which the topology was expanded.
- It is not countably compact, since the set [math]\displaystyle{ \{(\omega_1,-1/n) : n=2,3,\dots\} }[/math] has no upper limit point.
- It is not metacompact, since if [math]\displaystyle{ \{V_\alpha\} }[/math] is a covering of the ordinal space [math]\displaystyle{ [0,\omega_1) }[/math] with not point-finite refinement, then the covering [math]\displaystyle{ \{U_\alpha\} }[/math] of [math]\displaystyle{ X }[/math] defined by [math]\displaystyle{ U_1 = \{(0,\omega_1)\} \cup ([0,\omega_1] \times (0,1]), }[/math] [math]\displaystyle{ U_2 = [0,\omega_1] \times [-1,0), }[/math] and [math]\displaystyle{ U_\alpha = V_\alpha \times [-1,1] }[/math] has not point-finite refinement.
See also
- List of topologies – List of concrete topologies and topological spaces
References
- Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN:0-486-68735-X (Dover edition).
- S. Watson, The Construction of Topological Spaces. Recent Progress in General Topology, Elsevier, 1992.
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