Set-thoretic approach to the extension problem of continuous functions in topology

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The purpose of this research is to apply recent results of set-theory to the extension problem of a continuous function or a family of continuous functions on a topological space. In particular, we analyze the construction of Dowker spaces by Z. Balogh, which is one of key words in the problem. We give a combinatorial proof without using elementary submodels that Balogh's natural Dowker space is not countably paracompact (H. Ohta-T. Yorioka, 2012). Moreover, as by-products, we prove that every dense-in-itself separable metric space has an independent subbase (H. Ohta-H. Tsuiki-S. Yamada).