Vol. 82, No. 2, 1979

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Unicoherent plane Peano sets are σ-unicoherent

Wilfrido Martínez T. and Adalberto Garcia-Maynez Cervantes

Vol. 82 (1979), No. 2, 493–497

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 XMi is connected for every i, then X(M1M2) is connected.

Mathematical Subject Classification 2000
Primary: 54F55
Secondary: 54F25
Received: 15 September 1977
Revised: 15 December 1977
Published: 1 June 1979
Wilfrido Martínez T.
Adalberto Garcia-Maynez Cervantes