Addition theorem

for weights

If a Hausdorff compactum $X$ can be represented as the union over a set of infinite cardinality $\leq\tau$ of its subspaces of weight $\leq\tau$, then the weight of $X$ does not exceed $\tau$. The addition theorem (which was formulated as a problem in [1]) was established in [3] for $\tau=\aleph_0$ and in [4] in complete generality. Cf. Weight of a topological space.

References

[1]	P.S. Aleksandrov, P. Urysohn, "Mémoire sur les espaces topologiques compacts" , Koninkl. Nederl. Akad. Wetensch. , Amsterdam (1929)
[2]	R. Engelking, "General topology" , PWN (1977) (Translated from Polish)
[3]	Yu.M. Smirnov, "On metrizability of bicompacta, decomposable as a sum of sets with a countable base" Fund. Math. , 43 (1956) pp. 387–393 (In Russian)
[4]	A.V. Arkhangel'skii, "An addition theorem for weights of sets lying in bicompacta" Dokl. Akad. Nauk SSSR , 126 : 2 (1959) pp. 239–241 (In Russian)
[5]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)