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Keywords:
base; resolvable; partition
base; resolvable; partition
Summary:
We investigate whether an arbitrary base for a dense-in-itself topological space can be partitioned into two bases. We prove that every base for a $T_3$ Lindelöf topology can be partitioned into two bases while there exists a consistent example of a first-countable, 0-dimensional, Hausdorff space of size $2^\omega $ and weight $\omega_1$ which admits a point countable base without a partition to two bases.
References:
[1] Hajnal A., Hamburger P.: Set Theory. London Mathematical Society Student Texts, 48, Cambridge University Press, Cambridge, 1999, ISBN 0 521 59667 X. MR 1728582 | Zbl 0934.03057
[2] Stone A.H.: On partitioning ordered sets into cofinal subsets. Mathematika 15 (1968), 217–222. DOI 10.1112/S002557930000259X | MR 0237386 | Zbl 0164.33203
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