Abstract:
The paper investigates the structure of upper cones in the semilattice of Hausdorff compactifications of a topological space. For each compactification $\alpha A$ of $A$, there exists a unique perfect compactification $\gamma A$ that projects onto $\alpha A$ via a light map. This compactification $\gamma A$ is called the resolution (in the sense of Sklyarenko–Freudenthal) of $\alpha A$. Two characterizations of the resolution are given: it is the minimal one among perfect compactifications that project onto $\alpha A$, and the maximal one that projects onto $\alpha A$ via a light map. It is shown that the concept of resolution is a simultaneous generalization of the Freudenthal compactification and the unfolding of a manifold that is glued from polyhedra.
Additionally, properties of cluster sets of continuous mappings at points of a perfect extension of a given space are studied, and a sufficient condition for the existence of a continuous extension of a mapping is established. It is shown that any autohomeomorphism of the space that extends to an autohomeomorphism of some compactification, also extends to an autohomeomorphism of the resolution of this compactification.
Key words and phrases:perfect compactification, Freudenthal compactification, light map, cluster set.
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