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Ascoli's thoerems

Pasicki L.

Demonstratio Mathematica

2009

Vol. 42, nr 2

429-434

We present some elegant and pretty general versions of Ascoli theorem for non-Hausdorff spaces. The proofs are simple and elementary (e.g. we do not use the notion of joint continuity) while the results seem to be a very good starting point for more sophisticated theorems. Due to Lemma 10 (it extends [1], Th. 2.2) each of our theorems concerning even continuity is sufficiently general to contain the respective "equicontinuous" one (see Remark 18). Kelley in [2] (as regards the Ascoli theorems) was oriented mainly on continuous mappings while the present paper shows that it is natural first to get "continuous on compacta" results and t hen the "continuous1' versions become their simple consequences.