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Appert topology

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In general topology, a branch of mathematics, the Appert topology, named for Antoine Appert (1934), is a topology on the set X = {1, 2, 3, ...} of positive integers.[1] In the Appert topology, the open sets are those that do not contain 1, and those that asymptotically contain almost every positive integer. The space X with the Appert topology is called the Appert space.[1]

Construction

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For a subset S of X, let N(n,S) denote the number of elements of S which are less than or equal to n:

S is defined to be open in the Appert topology if either it does not contain 1 or if it has asymptotic density equal to 1, i.e., it satisfies

.

The empty set is open because it does not contain 1, and the whole set X is open since for all n.

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The Appert topology is closely related to the Fort space topology that arises from giving the set of integers greater than one the discrete topology, and then taking the point 1 as the point at infinity in a one point compactification of the space.[1] The Appert topology is finer than the Fort space topology, as any cofinite subset of X has asymptotic density equal to 1.

Properties

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  • The closed subsets S of X are those that either contain 1 or that have zero asymptotic density, namely .
  • Every point of X has a local basis of clopen sets, i.e., X is a zero-dimensional space.[1]
    Proof: Every open neighborhood of 1 is also closed. For any , is both closed and open.
  • X is Hausdorff and perfectly normal (T6).
    Proof: X is T1. Given any two disjoint closed sets A and B, at least one of them, say A, does not contain 1. A is then clopen and A and its complement are disjoint respective neighborhoods of A and B, which shows that X is normal and Hausdorff. Finally, any subset, in particular any closed subset, in a countable T1 space is a Gδ, so X is perfectly normal.
  • X is countable, but not first countable,[1] and hence not second countable and not metrizable.
  • A subset of X is compact if and only if it is finite. In particular, X is not locally compact, since there is no compact neighborhood of 1.
  • X is not countably compact.[1]
    Proof: The infinite set has zero asymptotic density, hence is closed in X. Each of its points is isolated. Since X contains an infinite closed discrete subset, it is not limit point compact, and therefore it is not countably compact.

See also

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Notes

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  1. ^ a b c d e f Steen & Seebach 1995, pp. 117–118

References

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  • Appert, Antoine (1934), Propriétés des Espaces Abstraits les Plus Généraux, Actual. Sci. Ind., Hermann, MR 3533016.
  • Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 0-486-68735-X.