Szymon Dolecki and David Gauld


Abstract. Regular and irregular pretopologies are studied. In particular,
for every ordinal there exists a topology such that the series of its partial
(pretopological) regularizations has length of that ordinal. Regularity and
topologicity of standard pretopologies on cascades can be characterized in
terms of their states, so that their study for such spaces reduces to that of
a combinatorics of states. For example, if an iterated partial regularization
r^kpi is topological for k > 0 then rpi is a regular topology. Irregularity of
pretopologies of countable character can be characterized in terms of sequential
cascades with standard irregular pretopologies.

convergence spaces, pretopologies, partial regularisation, cascades

Math Review Classification
Primary 54A20, 54D10

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26 pages

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