Applied Categorical Structures

, Volume 22, Issue 3, pp 551–563

Function Spaces and Contractive Extensions in Approach Theory: The Role of Regularity


  • Eva Colebunders
    • Vakgroep WiskundeVrije Universiteit Brussel
    • Mathematical SciencesGeorgia Southern University
  • William Trott
    • Mathematical SciencesGeorgia Southern University

DOI: 10.1007/s10485-013-9321-z

Cite this article as:
Colebunders, E., Mynard, F. & Trott, W. Appl Categor Struct (2014) 22: 551. doi:10.1007/s10485-013-9321-z


Two classical results characterizing regularity of a convergence space in terms of continuous extensions of maps on one hand, and in terms of continuity of limits for the continuous convergence on the other, are extended to convergence-approach spaces. Characterizations are obtained for two alternative extensions of regularity to convergence-approach spaces: regularity and strong regularity. The results improve upon what is known even in the convergence case. On the way, a new notion of strictness for convergence-approach spaces is introduced.


Regularity Strong regularity Convergence space Convergence-approach space Approach space Strict subspace Continuous extension Contractive extension Default of contraction Continuous convergence Diagonal axioms

Mathematics Subject Classifications (2010)

54A20 54A05 54D10 54B30 54B05 54C05 54C35

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© Springer Science+Business Media Dordrecht 2013