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The sequential topology on is not regular

Published online by Cambridge University Press:  08 September 2009

MATTHIAS SCHRÖDER
MATTHIAS SCHRÖDER*
Affiliation:
Fakultät für Informatik, Universität der Bundeswehr München, 85577 Neubiberg, Germany Email: [email protected]
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Abstract

The compact-open topology on the set of continuous functionals from the Baire space to the natural numbers is well known to be zero-dimensional. We prove that the closely related sequential topology on this set is not even regular. The sequential topology arises naturally as the topology carried by the exponential formed in various cartesian closed categories of topological spaces. Moreover, we give an example of an effectively open subset of that violates regularity. The topological properties of are known to be closely related to an open problem in Computable Analysis. We also show that the sequential topology on the space of continuous real-valued functions on a Polish space need not be regular.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 19 , Issue 5 , October 2009 , pp. 943 - 957
Copyright
Copyright © Cambridge University Press 2009

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