Unicoherent plane Peano sets are σ-unicoherent

Wilfrido Martínez T. and Adalberto Garcia-Maynez Cervantes

Vol. 82 (1979), No. 2, 493–497

DOI: 10.2140/pjm.1979.82.493

Abstract

A Peano space is a connected, locally connected and locally compact metric space. A region in a space X is an open and connected subset of X. A space X is σ-eonnected if every sequence A₁,A₂,⋯ of closed, mutually disjoint subsets of X, with at least two of them nonempty, fails to cover X. A connected space X is unicoherent (resp., σ-unicoherent) if for every pair H, K of closed and connected (resp., and σ-connected) sets with union X, the intersection H ∩ K is connected (resp., σ-connected).

Theorem. Let X be a plane Peano space. Then the following properties are equivalent:

1. X is unicoherent;
2. There exists a cover of X formed by unicoherent regions U₁ ⊂ U₂ ⊂ ⋯ with compact closures;
3. X is σ-unicoherent, and
4. If M₁,M₂,⋯ is a sequence of closed, mutually disjoint subsets of X such that X−M_i is connected for every i, then X−(M₁∪M₂∪⋯) is connected.

Mathematical Subject Classification 2000

Primary: 54F55

Secondary: 54F25

Milestones

Received: 15 September 1977

Revised: 15 December 1977

Published: 1 June 1979