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Dugundji compact

Last-modified: 2010-12-01 (水) 01:43:34

Definition Edit

  • a compact space X is Dugundji if it has one of the following equivalent properties:
    1. if Z is a zero-dimensional compact space and A is a closed subset of Z, then
      every continuous map A → X extends to a continuous map Z → X;
    2. there exists a family Φ of closed equivalence relations on X such that:
      1. for every imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20R%5cin%5cPhi%20%5c%5d%7d%25.png the quotient space X/R is metrizable and the quotient map X → X/R is open;
      2. ∩Φ = the diagonal of X^2 (in other words, the family Φ separates points of X);
      3. Φ is closed under countable intersections.

Property Edit

  • For every topological group X every compact G_δ-subset of X is Dugundji.
  • For every Mul’tsetv space? X every compact G_δ-subspace of X is Dugundji.

Reference Edit

  • Reznichenko, E. A.(RS-MOSCM-GT) and Uspenskij, V. V.(RS-MOSCM-GT), Pseudocompact Malʹtsev spaces. (English summary) ,Special issue on topological groups. ,Topology Appl. 86 (1998), no. 1, 83--104.