用诱导映射表征紧凑 Hausdorff 空间的开映射和半开映射

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2024-04-23 DOI:10.1016/j.topol.2024.108921

Xiongping Dai, Yuxuan Xie

引用次数: 0

摘要

设 f:X→Y 是紧凑 Hausdorff 空间的连续投射。用f⁎:M(X)→M(Y),μ↦μ∘f-1和2f:2X→2Y,A↦f[A]分别表示概率度量空间和超空间上的诱导连续投射。本文主要证明以下事实：(1)若 f⁎ 是半开的，则 f 是半开的；(2)若 f 是半开的密开的，则 f⁎ 是半开的密开的；(3)若 2f 是开的，则 f 是开的；(4)若 2f 是半开的，则 f 是半开的；(5)若 2f 是不可还原的，则 f 是不可还原的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Characterizations of open and semi-open maps of compact Hausdorff spaces by induced maps

Let $f : X \to Y$ be a continuous surjection of compact Hausdorff spaces. By $f_{⁎} : M (X) \to M (Y), μ \mapsto μ \circ f^{- 1} and 2^{f} : 2^{X} \to 2^{Y}, A \mapsto f [A]$ we denote the induced continuous surjections on the probability measure spaces and hyperspaces, respectively. In this paper we mainly show the following facts:

(1)
If $f_{⁎}$ is semi-open, then f is semi-open.
(2)
If f is semi-open densely open, then $f_{⁎}$ is semi-open densely open.
(3)
f is open iff $2^{f}$ is open.
(4)
f is semi-open iff $2^{f}$ is semi-open.
(5)
f is irreducible iff $2^{f}$ is irreducible.

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来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.