Sometimes it happens that you have a sequence of topological spaces each contained in the next $$ X_1 \subset X_2 \subset X_3 \subset \ldots$$ and you want to talk about things like homotopy in the union $$X = \bigcup_{n=1}^\infty X_n$$ the idea being that, as $n \to \infty$, you have a bit more space to move things around.
Now, there are natural ways of topologizing this union. For example, $X$ can be topologized so that a map out of $X$ is continuous if and only if its restriction to each $X_n$ is continuous.
But often all you really want is to do your deformations in some finite stage of the union. For example, you might want to call $x, y \in X$ asymptotically path equivalent if, for some $n$, $x$ and $y$ are joined by a path in $X_n$ (hence in $X_k$ as well for all $k \geq n$).
I have two questions:
- Is there standard terminology for such notions?
- Is there a topology on $X$ which captures these notions automatically? For example, if it could be arranged so that each $X_n$ is open in $X$ (which may be impossible since they might not be open in each other) then any path $[0,1] \to X$ would have to remain in some $X_n$ by a compactness argument.