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Connectivity degrees of complements of closed sets in continua

Mauricio Chacón-Tirado, César Piceno

Abstract

In the literature, various types of points and meager sets whose complements are connected have been studied, such as colocally connected points, non-weak cut points/sets, non-block points/sets, shore points/sets, etc. We extend that study, in the following way: considering a continuum $X$ and a natural number $n$, we investigate sets $A \in 2^X$ meeting the criterion that $X - A$ has at most $n$ components, and we introduce degrees of connectivity of the complement of $A$. When $n=1$ and $A$ is meager or singleton, these new definitions are equivalent to the known definitions of non-cut points/sets.

Connectivity degrees of complements of closed sets in continua

Abstract

In the literature, various types of points and meager sets whose complements are connected have been studied, such as colocally connected points, non-weak cut points/sets, non-block points/sets, shore points/sets, etc. We extend that study, in the following way: considering a continuum and a natural number , we investigate sets meeting the criterion that has at most components, and we introduce degrees of connectivity of the complement of . When and is meager or singleton, these new definitions are equivalent to the known definitions of non-cut points/sets.
Paper Structure (5 sections, 37 theorems, 6 equations, 1 figure)

This paper contains 5 sections, 37 theorems, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. General properties of non-$n$-cut sets
  4. Properties of non-$n$-cut sets preserved by continuous functions
  5. Relations between $X$ and the hyperspaces of non-cut sets of degree $n$ of $X$

Key Result

Theorem 3.1

Given a continuum $X$ and $n \in \mathbb N$, the following conditions hold:

Figures (1)

  • Figure 1: Relationships between degrees of connectivity.

Theorems & Definitions (85)

  • Definition 2.1
  • Definition 2.3
  • Theorem 3.1
  • Example 3.2
  • Definition 3.5
  • Theorem 3.6
  • proof
  • Lemma 3.7
  • proof
  • Theorem 3.8
  • ...and 75 more