Recall that the Singular Cardinals Hypothesis (SCH) says that if $\kappa$ is a singular cardinal and $2^{cf(\kappa)}<\kappa,$ then $\kappa^{cf(\kappa)}=\kappa^+.$ Clearly it has many applications in set theory, for example assuming SCH, the behavior of the power function is determined by its behavior on regular cardinals, ....

What are the main applications of SCH outside of set theory, in particular in model theory, algebra, analysis, topology, and ...(with references)?

Remark. One application that I am aware is given in the following: "Cater, F. S.; Erdős, Paul; Galvin, Fred On the density of λ-box products. General Topology Appl. 9 (1978), no. 3, 307–312."

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    $\begingroup$ In the paper "A topological equivalence of the singular cardinals hypothesis", it is proved that SCH is equivalent to the following: Each metric space of cardinality greater than the continuum and of weight of uncountable cofinality has cardinality equal to its weight. An equivalent condition for a larger class of topological spaces is also presented. See also the following papers: 1) A topological reflection principle equivalent to Shelah’s strong hypothesis, Assaf Rinot 2) Openly generated Boolean algebras and the Fodor-type reflection principle, Assaf Rinot, Sakaé Fuchino. $\endgroup$ – Mohammad Golshani Apr 12 '14 at 3:37

Application in group theory:

Galindo, Jorge, and Sergio Macario. "Pseudocompact group topologies with no infinite compact subsets." Journal of Pure and Applied Algebra 215.4 (2011): 655-663. arxiv


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