Compactness theorem (English Wikipedia)

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J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985) [1], in particular, Makowsky, J. A. Chapter XVIII: Compactness, Embeddings and Definability. 645--716, see Theorems 4.5.9, 4.6.12 and Proposition 4.6.9. For compact logics for an extended notion of model see Ziegler, M. Chapter XV: Topological Model Theory. 557--577. For logics without the relativization property it is possible to have simultaneously compactness and interpolation, while the problem is still open for logics with relativization. See Xavier Caicedo, A Simple Solution to Friedman's Fourth Problem, J. Symbolic Logic, Volume 51, Issue 3 (1986), 778-784.doi:10.2307/2274031 JSTOR 2274031

J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985) [1], in particular, Makowsky, J. A. Chapter XVIII: Compactness, Embeddings and Definability. 645--716, see Theorems 4.5.9, 4.6.12 and Proposition 4.6.9. For compact logics for an extended notion of model see Ziegler, M. Chapter XV: Topological Model Theory. 557--577. For logics without the relativization property it is possible to have simultaneously compactness and interpolation, while the problem is still open for logics with relativization. See Xavier Caicedo, A Simple Solution to Friedman's Fourth Problem, J. Symbolic Logic, Volume 51, Issue 3 (1986), 778-784.doi:10.2307/2274031 JSTOR 2274031

J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985) [1], in particular, Makowsky, J. A. Chapter XVIII: Compactness, Embeddings and Definability. 645--716, see Theorems 4.5.9, 4.6.12 and Proposition 4.6.9. For compact logics for an extended notion of model see Ziegler, M. Chapter XV: Topological Model Theory. 557--577. For logics without the relativization property it is possible to have simultaneously compactness and interpolation, while the problem is still open for logics with relativization. See Xavier Caicedo, A Simple Solution to Friedman's Fourth Problem, J. Symbolic Logic, Volume 51, Issue 3 (1986), 778-784.doi:10.2307/2274031 JSTOR 2274031

Marker 2002, pp. 40–43. Marker, David (2002). Model Theory: An Introduction. Graduate Texts in Mathematics. Vol. 217. Springer. ISBN 978-0-387-98760-6. OCLC 49326991.
Gowers, Barrow-Green & Leader 2008, pp. 639–643. Gowers, Timothy; Barrow-Green, June; Leader, Imre (2008). The Princeton Companion to Mathematics. Princeton: Princeton University Press. pp. 635–646. ISBN 978-1-4008-3039-8. OCLC 659590835.