Since closed subsets of compact spaces are compact, every set in a compact space is relatively compact. In the case of a metric topology, or more generally when sequences may be used to test for compactness, the criterion for relative compactness becomes that any sequence in Y has a subsequence convergent in X. This condition is also called pre-compact or relatively bounded.
Some major theorems characterise relatively compact subsets, in particular in function spaces. An example is the Arzela-Ascoli theorem. Other cases of interest relate to uniform integrability, and the concept of normal family in complex analysis. Mahler's compactness theorem in the geometry of numbers characterises relatively compact subsets in certain non-compact homogeneous spaces (specifically spaces of lattices).
The definition of almost periodic functionF is at a conceptual level to do with the translates of F being a relatively compact set. This needs to be made precise in terms of the topology used, in a particular theory.
The name of the Shakhbazian compact group of galaxies as constructed from the sequence number (listed in the original references and in the originating CDS Catalog) combined with the prefix `SHK' which is recommended by the Dictionary of Astronomical Nomenclature.
The number of galaxies in the compact group: in the published lists, this number was frequently stated as a range, notice, and these ranges were apparently converted (by the authors) to their median values when the machine-readable version was created.
The coefficient of relativecompactness of the group of galaxies: this coefficient is the ratio of the sum of the diameters of all the galaxies of the group to the diameter of the group as a whole.
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