Every finite system of T1 uniformities comes from a single distance structure


  • Jobst Heitzig Universität Hannover




Distance function, Free monoid, Generalized metric, Uniformity


Using the general notion of distance function introduced in an earlier paper, a construction of the finest distance structure which induces a given quasi-uniformity is given. Moreover, when the usual defining condition xy : d(y; x)  of the basic entourages is generalized to nd(y; x)    n (for a  fixed positive integer n), it turns out that if the value-monoid of the distance function is commutative, one gets a countably infinite family of quasi-uniformities on the underlying set. It is then shown that at least every finite system and every descending sequence of T1 quasi-uniformities which fulfil a weak symmetry condition is included in such a family. This is only possible since, in contrast to real metric spaces, the distance function need not be symmetric.


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Author Biography

Jobst Heitzig, Universität Hannover

Institut für Mathematik

Universität Hannover

Welfengarten 1

D-30167 Hannover



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How to Cite

J. Heitzig, “Every finite system of T1 uniformities comes from a single distance structure”, Appl. Gen. Topol., vol. 3, no. 1, pp. 65–76, Apr. 2002.



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