Example

Let $I$ be a directed set and $(X_\alpha)_{\alpha \in I}$ a family of subsets of a set $Y$, such that $X_\alpha \supset X_\beta$ whenever $\alpha \leq \beta$. For each $\alpha \in I$ let $\mathfrak{T}_\alpha$ be a topology on $X_\alpha$ such that $\mathfrak{T}_\beta$ is finer than the topology induced on $X_\beta$ by $\mathfrak{T}_\alpha$ whenever $\alpha \leq \beta$. If we take $f_{\alpha\beta}$ to be the canonical injection $X_\beta \to X_\alpha$ for $\alpha \leq \beta$, then $\lim X_\alpha$ may be identified canonically with the intersection $X$ of the $X_\alpha$, with the topology which is the least upper bound (§ 2, no. 3, Example 2) of the topologies induced on $X$ by the $\mathfrak{T}_\alpha$.