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Compactifications, realcompletions and spectral structures

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Summary

We give an intrinsic (set theoretical) method to obtain all realcompletions of a Tychonoff space X. It is based on the concept of a spectral structure. Every realcompletion of the space can be obtained as a space of terminal clusters relative to an appropriate spectral structure on X. Various applications of this concept are then given. For example we may characterize those spectral structures which yield the realcompletions between X and βX, or which yield spaces that are realcomplete (that is, realcompact), or compact or pseudocompact or Lindelöf. We also determine the class K of compactifications K of X for which X will be real closed in K for every K ε K.

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Additional information

Here we continue the use of the word realcompletion (see [2]) in place of real-compactification. As many investigations have shown and as the present article reinforces, the former is more appropriate than the latter.

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Alò, R.A., Talamo, R. Compactifications, realcompletions and spectral structures. Annali di Matematica pura ed applicata 124, 127–137 (1980). https://doi.org/10.1007/BF01795389

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Keywords

  • Yield Space
  • Spectral Structure
  • Tychonoff Space
  • Terminal Cluster