Nonstandard homology theory for uniform spaces

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• 作者：Takuma Imamura ^{s1240008@ems.u-toyama.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：55N35 ; 54J05
• 刊名：Topology and its Applications
• 出版年：2016
• 出版时间：15 August 2016
• 年：2016
• 卷：209
• 期：Complete
• 页码：22-29
• 全文大小：283 K

文摘

We introduce a new homology theory of uniform spaces, provisionally called μ-homology theory. Our homology theory is based on hyperfinite chains of microsimplices. This idea is due to McCord. We prove that μ-homology theory satisfies the Eilenberg–Steenrod axioms. The characterization of chain-connectedness in terms of μ-homology is provided. We also introduce the notion of S-homotopy, which is weaker than uniform homotopy. We prove that μ-homology theory satisfies the S-homotopy axiom, and that every uniform space can be S-deformation retracted to a dense subset. It follows that for every uniform space X and any dense subset A of X, X and A have the same μ-homology. We briefly discuss the difference and similarity between μ-homology and McCord homology.