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<p>Let <i>X</i> be a topological space, let ∆ be a partition of <i>X</i>, and let <i>Y</i> = <i>X</i>/∆ be a quotient space with the corresponding quotient topology. Then the automorphism group <i>ℋ</i>(∆) of ∆ (i.e., the homeomorphisms of <i>X</i> that permute the elements of partition) acts in a natural way upon <i>Y</i> by homeomorphisms. We determine the cases in which the corresponding homomorphism of the action <i>ψ</i>: <i>ℋ</i>(∆) <i>→ ℋ</i>(<i>Y</i>) into the group of homeomorphisms of <i>Y</i> is continuous with respect to the compact-open topologies. The obtained results have applications to the foliation theory.</p>

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Automorphisms and Endomorphisms of the Partitions of Topological Spaces

  • 期刊论文

  • 发表日期：

    2026年5月22日

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  • Sergiy Maksymenko,
  • Eugene Polulyakh

摘要

Let X be a topological space, let ∆ be a partition of X, and let Y = X/∆ be a quotient space with the corresponding quotient topology. Then the automorphism group ℋ(∆) of ∆ (i.e., the homeomorphisms of X that permute the elements of partition) acts in a natural way upon Y by homeomorphisms. We determine the cases in which the corresponding homomorphism of the action ψ: ℋ(∆) → ℋ(Y) into the group of homeomorphisms of Y is continuous with respect to the compact-open topologies. The obtained results have applications to the foliation theory.