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