Let $X$ be a set. A partition of $X$ is a collection of subsets $\{E_\alpha\}\subset\mathcal{P}(X)$ such that the following holds.
- For every $x\in X$, there is some $\alpha$ such that $x\in E_\alpha$.
- For each $\alpha$ and $\alpha'$, $E_\alpha\cap E_{\alpha'}=\emptyset$.
Is there a name for a collection which is only required to satisfy the first condition? If not, suggestions for a name I might use? I am currently thinking "surjective collection" or "covering collection" or "semi-partition", but am not very satisfied with any of these.
As far as I've seen, the word cover is usually employed for this (no need for extra adjectives).
That is, $\{E_\alpha\}_{\alpha\in A}\subset\mathcal{P}(X)$ is a cover of $X$ precisely when $X=\bigcup\limits_{\alpha\in A}E_\alpha$.