does every topology have a basis?

This might be a silly question, but i was wondering, is there any topology that cannot be generated by a basis? if not, given a topology, is there a reliable way of figuring out a basis for it? it probably matters if the set $X$ the topology is on is countable or not, right? Would it matter if the topology itself is countable? Thanks for any help/advice/feedback.

Every topology is a base for itself.

Added: On any set $X$ the indiscrete topology $\{X,\varnothing\}$ has only itself as base. There are other examples. For instance, for $n\in\Bbb Z^+$ let $V_n=\{1,\dots,n\}$, and let $\tau=\{\varnothing,\Bbb N\}\cup\{V_n:n\in\Bbb Z^+\}$; then $\tau$ is a topology on $\Bbb N$ whose only base is itself: none of the sets $V_n$ can be written as a union of the other sets in the topology, and $\Bbb N$ is the only open set containing $0$.

However, if each point of the space $\langle X,\tau\rangle$ has an open nbhd that is not the whole space, then $\tau\setminus\{X\}$ is always a base of $\tau$ different from $\tau$ itself. In particular this is true of every $T_1$ space with more than one point.