This article has multiple issues. Please help or discuss these issues on the talk page. (Learn how and when to remove these template messages)
This article needs additional citations for verification. Please help by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: – ···scholar·JSTOR(January 2019) (Learn how and when to remove this template message)
The topic of this article may not meet Wikipedia's notability guideline for numbers. Please help to demonstrate the notability of the topic by citing reliable secondary sources that are independent of the topic and provide significant coverage of it beyond a mere trivial mention. If notability cannot be shown, the article is likely to be merged, redirected, or deleted. Find sources: – ···scholar·JSTOR(January 2019) (Learn how and when to remove this template message)
(Learn how and when to remove this template message)
The construction of the Alexandroff plank starts by defining the topological space to be the Cartesian product of and , where is the first uncountable ordinal, and both carry the interval topology. The topology is extended to a topology by adding the sets of the form
where .
The Alexandroff plank is the topological space .
It is called plank for being constructed from a subspace of the product of two spaces.
Properties[]
The space satisfies that:
is Urysohn, since is regular. The space is not regular, since is a closed set not containing , while every neighbourhood of intersects every neighbourhood of .
is semiregular, since each basis rectangle in the topology is a regular open set and so are the sets defined above with which the topology was expanded.
is not countably compact, since the set has no upper limit point.
is not metacompact, since if is a covering of the ordinal space with not point-finite refinement, then the covering of defined by , , and has not point-finite refinement.
References[]
Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN0-486-68735-X (Dover edition).
S. Watson, The Construction of Topological Spaces. Recent Progress in General Topology, Elsevier, 1992.
This topology-related article is a stub. You can help Wikipedia by .
v
t
Categories:
Topological spaces
Topology stubs
Hidden categories:
Articles needing additional references from January 2019
All articles needing additional references
Articles with topics of unclear notability from January 2019