Abstract \(\omega\)-limit sets. (English) Zbl 1406.54020
A function \(f: X\to X\) is called weakly incompressible if for any proper and nontrivial open set \(U\subset X\) the image \(f(cl_X (U))\) is not contained in \( U\). The shift map \(\sigma: \beta\omega \to \beta \omega\), where \(\sigma(p) = \{V+1: V\in p \}\), and its restriction to \(\omega^*\) (recall that \(\omega^*\) consists of all free ultrafilters on \(\omega\)) are weakly incompressible. If a function \(f:X\to X\) is continuous, then the pair \((X,f)\) is called a quotient of \((\omega^*, \sigma)\) whenever there exists a continuous surjection \(Q: \omega^* \to X\) such that
\[
Q \circ f = f\circ Q.
\]
The main theorem says:
Suppose \(X\) is a compact Hausdorff space with weight at most \(\aleph_1\). Then, a continuous function \(f:X \to X\) is weakly incompressible iff the pair \((X,f)\) is a quotient of \((\omega^*, \sigma)\).
Suppose \(X\) is a compact Hausdorff space with weight at most \(\aleph_1\). Then, a continuous function \(f:X \to X\) is weakly incompressible iff the pair \((X,f)\) is a quotient of \((\omega^*, \sigma)\).
Reviewer: Szymon Plewik (Katowice)
MSC:
| 54H20 | Topological dynamics (MSC2010) |
| 03C98 | Applications of model theory |
| 03E35 | Consistency and independence results |
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
| 54G05 | Extremally disconnected spaces, \(F\)-spaces, etc. |
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