Definition of the Space $\mathbb{R}^\infty$ I'm trying to understand the definition of the topological space $\mathbb{R}^{\infty}$, which Hatcher defines as $\cup_n \mathbb{R}^n$ in his book Algebraic Topology. I'm having trouble making sense of this union since $\mathbb{R}^m$ is not literally a subset of $\mathbb{R}^n$ for $m < n$. Certainly the first embeds into the second, but in order to form the space $\bigcup_n \mathbb{R}^n$ and give it the weak topology we need an infinite chain of subsets $\mathbb{R}^0 \subset \mathbb{R}^1 \subset \mathbb{R}^2 \subset \cdots$. We can almost obtain this by replacing each Euclidean space by its embedded image in higher-dimensional Euclidean spaces, but this approach falls short because it will only give us finite subset chains. How do we get around this formal problem?

I think this is just a common shorthand in algebraic topology. More rigorously, I think he's defining $\mathbb{R}^\infty$ as the colimit of the diagram $\mathbb{R}^1\hookrightarrow\mathbb{R}^2\hookrightarrow\dots$ I believe the topology this colimit inherits from each Euclidean space will be the one you're thinking of, but I could be wrong.
