Let $(\Lambda, \le)$ be a directed system and $\{ X_{\alpha} \}_{\alpha \in \Lambda}$ be a family of topological spaces indexed by $\Lambda$ such that $X_{\alpha} \subseteq X_{\beta}$ whenever $\alpha \le \beta$, and the inclusion map $i_{\alpha \beta} : X_{\alpha} \hookrightarrow X_{\beta}$ is continuous. Let $X := \bigcup_{\alpha \in \Lambda} X_{\alpha}$.
I want the following notion of convergence on $X$: A net $(x_i)$ in $X$ is said to converge to $x \in X$ if and only if there is an index $\alpha \in \Lambda$ such that $(x_i)$ is eventually contained in $X_{\alpha}$, $x \in X_{\alpha}$, and $x_i \rightarrow x$ in $X_{\alpha}$.
When $\Lambda$ is countable, there is the idea of strict inductive limits which is, for instance, useful in topologizing the space of smooth functions on $\mathbb{R}^n$ with compact support. For $\Lambda$ of arbitrary cardinality, this notion of convergence seems well-defined to me although it is quite possible that it is not compatible with any idea of topological convergence.
Question: What is the appropriate framework that would describe the above notion of convergence in $X$? It is not clear to me that even the framework of convergence spaces is appropriate here.
Thank you.
