# Property of a Collection of Subsets Containing the Base Set

The definition of a topology $\mathcal{T}$ on a set $X$ requires several things, one of which is that the set $X$ is an element of $\mathcal{T}$. Similarly, one of the ingredients of the definition of a $\sigma$-algebra $\mathcal{A}$ on a set $X$ is that $X \in \mathcal{A}$. In each case, we are defining a collection of subsets of a set $X$ that satisfy certain criteria and a common requirement is that the set $X$, itself, be a part of the constructed collection. Is there a general term for this requirement that could be applied in both instances? That is, given a collection $\mathcal{C}(X)$ of subsets of $X$ the property that $X \in \mathcal{C}(X)$ is called _____ ?

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I would call it a $\mathcal{C}$-set without further information. This is a bit off-topic, but I was reminded of the Borel-Hierarchy: Open sets are called $G$-sets ($G$: German Gebiet - region), closed sets $F$-sets ($F$ French fermé - closed), countable intersections of open sets $G_\delta$-sets, countable unions of closed sets $F_\sigma$ sets, countable unions of $G_\delta$-sets are $G_{\delta\sigma}$ sets, etc (each letter $\sigma$, $\delta$ corresponds to taking countable unions/intersections). This goes back to Hausdorff (at least). –  t.b. Nov 19 '11 at 0:55
@t.b. I wasn't aware of these classifications but can see how they might be useful. –  ItsNotObvious Nov 19 '11 at 1:41