“Arens–Fort space”的意思、由来-开放百科全书

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Arens–Fort space

释义

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In mathematics, the Arens–Fort space is a special example in the theory of topological spaces, named for Richard Friederich Arens and M. K. Fort, Jr.

Let X be a set of ordered pairs of non-negative integers (m, n). A subset U of X is open if and only if:

it does not contain (0, 0), or
it contains (0, 0), and all but a finite number of points of all but a finite number of columns, where a column is a set {(m, n)} with fixed m.

In other words, an open set is only "allowed" to contain (0, 0) if only a finite number of its columns contain significant gaps. By a significant gap in a column we mean the omission of an infinite number of points.

It is

Hausdorff
regular
normal

It is not:

second-countable
first-countable
metrizable
compact

References

{{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=Counterexamples in Topology | origyear=1978 | publisher=Springer-Verlag | location=Berlin, New York | edition=Dover reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 | year=1995}}

1 : Topological spaces

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