Geodesic
A generalization of the Helly theorem for functions with values in a~uniform space
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2010), pp. 41-54

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In this paper we consider sequences of functions that are defined on a subset of the real line with values in a uniform Hausdorff space. For such sequences we obtain a sufficient condition for the existence of pointwise convergent subsequences. We prove that this generalization of the Helly theorem includes many results of the recent research. In addition, we prove that the sufficient condition is also necessary for uniformly convergent sequences of functions. We also obtain a representation for regular functions whose values belong to the uniform space.
Keywords: selection principle, pointwise convergence, proper functions with respect to a dense set, uniform space. 
@article{IVM_2010_5_a5,
     author = {Yu. V. Tret'yachenko},
     title = {A generalization of the {Helly} theorem for functions with values in a~uniform space},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {41--54},
     publisher = {mathdoc},
     number = {5},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2010_5_a5/}
}
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Yu. V. Tret'yachenko. A generalization of the Helly theorem for functions with values in a~uniform space. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2010), pp. 41-54. http://geodesic.mathdoc.fr/item/IVM_2010_5_a5/