For Hausdorff topological monoids, the concept of a unitary Cauchy net is a generalization of the concept of a fundamental sequence of reals. We consider properties and applications of such nets and of corresponding filters and prove, in particular, that the underlying set of a given monoid, endowed with the family of such filters, forms a Cauchy space whose convergence structure defines a uniform topology. A commutative monoid endowed with the corresponding uniformity is uniform. A distant purpose of the paper is to transfer the classical concepts of a completeness and of a completion into the theory of topological monoids.
@article{bwmeta1.element.doi-10_2478_taa-2013-0006,
author = {Boris G. Averbukh},
title = {On unitary Cauchy filters on topological monoids},
journal = {Topological Algebra and its Applications},
volume = {1},
year = {2013},
pages = {46-59},
zbl = {1288.22001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_taa-2013-0006}
}
Boris G. Averbukh. On unitary Cauchy filters on topological monoids. Topological Algebra and its Applications, Tome 1 (2013) pp. 46-59. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_taa-2013-0006/
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