TITLE: cardinality of a basis for a topology
QUESTION [8 upvotes]: Suppose X is a space of cardinality $\le \kappa$. I would like to claim that any topology on X has a basis of cardinality $\le \kappa$. Intuitively it's true since even the discrete topology has such basis but i can't prove or find a counter-example...
Thanks!

REPLY [6 votes]: It does not appear to be true. Let me recommend a wonderful website $\pi$-Base where you can search for topologies having certain properties. In particular I searched for a countable space which is not second countable (no countable base) here. It gives various examples.