On wikipedia, it says that every covering space is a fiber bundle. My understanding of a fiber bundle is that you have a topological space $F$ and $\pi:Y\to X$ surjective and for every $x\in X$, $\pi^{-1}(x)$ is homeomorphic to $F$. So each fiber has the same cardinality.
Now I know that if $X$ is connected, then every fiber has the same cardinality, but is this true for general $X$?
It is not true, since you could just take separate covering maps over disconnected parts which have fibers of different size. For a really simple example, if $X$ and $Y$ are discrete spaces then any map $Y\to X$ is a covering map but the fibers of an arbitrary map certainly do not need to have the same cardinality.
(Here I do not require covering maps to be surjective, which is the better definition, but in any case the fibers of a surjective map do not have to have the same cardinality either.)