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There's been quite a bit of discussion of this thread in the meta thread. I'd like to take a stab at answering my interpretation of what the question is getting at. In a sense there's at least two things going on, and that's part of why I think there's been so much discussion.

To take a step back from Grassmannians, I'd like to mention why one might be interested in $\mathbb R^\infty$. The Whitney embedding theorem states that every continuous function $f : N \to \mathbb R^k$ can be approximated uniformly by a $C^\infty$-smooth embedding provided $k \geq 2n+1$ where the dimension of $N$ is $n$. Moreover, any two embeddings $N \to \mathbb R^k$ are isotopic provided $k \geq 2n + 2$. So if you wanted to, you could replace the class "$n$-manifolds up to diffeomorphism" with "isotopy classes of $n$-dimensional submanifolds of $\mathbb R^k$ provided $k \geq 2n+2$".

A key nice result about the weak topology on $\mathbb R^\infty$ is that any continuous function from a compact space to $\mathbb R^\infty$ has an image in $\mathbb R^k$ for some $k$. So from the perspective of the Whitney embedding theorem above, "$n$-dimensional manifolds up to diffeomorphism" is precisely "$n$-dimensional submanifolds of $\mathbb R^\infty$ up to isotopy". The key thing here is the ambient space is now independent of the dimension of the manifold you're talking about. This is pretty much exactly what's going on with the Grassmannians.

Given a vector bundle $p : E \to B$ over a finite-dimensional space $B$ (say a manifold or a CW-complex), there exists a classifying map for the bundle, meaning $p$ is isomorphic to the pull-back of the tautological bundle over $G_{n,k} \equiv G_k(\mathbb R^n)$. $n$ is just some sufficiently large integer. Although people don't state it this way, this theorem is basically the Whitney embedding theorem but for vector bundles. Because it's saying that up to isomorphism, $E$ is a collection of pairs $(b,v)$ where $b \in B$ and $v \in \chi^{-1}(h(b))$, where $\chi : E_{n,k} \to G_{n,k}$ is the tautological bundle over $G_{n,k}$. $h : B \to G_{n,k}$ the classifying map. In a sense we've "embedded" the vector bundle in Euclidean space, well, we've made the fibers as subspaces of Euclidean space.

But again, you have the "for some $n$ sufficiently large" thing. And like with manifolds $n$ has an upper bound in terms of the dimension of $B$ (if $B$ is finite dimensional). Since it's sometimes awkward to carry-around these "for sufficiently large $n$" statements, you take the limit space $G_{\infty,k}$ and now your statement is far more clean, because any $k$-dimensional vector bundle over any space $B$ is the pull back of some map $B \to G_{\infty,k}$. The point is that $G_{\infty,k}$ is a universal space -- independent of $B$ or the vector bundle over $B$. The weak topology is exactly what allows us to ensure this happens.

show/hide this revision's text 2 added 73 characters in body

There's been quite a bit of discussion of this thread in the meta thread. I'd like to take a stab at answering my interpretation of what the question is getting at. In a sense there's at least two things going on, and that's part of why I think there's been so much discussion.

To take a step back from Grassmannians, I'd like to mention why one might be interested in $\mathbb R^\infty$. The Whitney embedding theorem states that every continuous function $f : N \to \mathbb R^k$ can be approximated uniformly by a $C^\infty$-smooth embedding provided $k \geq 2n+1$ where the dimension of $N$ is $n$. Moreover, any two embeddings $N \to \mathbb R^k$ are isotopic provided $k \geq 2n + 2$. So if you wanted to, you could replace the class "$n$-manifolds up to diffeomorphism" with "isotopy classes of $n$-dimensional submanifolds of $\mathbb R^k$ provided $k \geq 2n+2$".

A key nice result about the weak topology on $\mathbb R^\infty$ is that any continuous function from a compact space to $\mathbb R^\infty$ has an image in some $\mathbb R^k$ for some $k$. So from the perspective of the Whitney embedding theorem above, "$n$-dimensional manifolds up to diffeomorphism" is precisely "$n$-dimensional submanifolds of $\mathbb R^\infty$ up to isotopy". The key thing here is the ambient space is now independent of the dimension of the manifold you're talking about. This is pretty much exactly what's going on with the Grassmannians.

Given a vector bundle $p : E \to B$ over a paracompact finite-dimensional space $B$, B$ (say a manifold or a CW-complex), there exists a classifying map for the bundle, meaning $p$ is isomorphic to the pull-back of the tautological bundle over $G_{n,k} \equiv G_k(\mathbb R^n)$. $n$ is just some sufficiently large integer. Although people don't state it this way, this theorem is basically the Whitney embedding theorem but for vector bundles. Because it's saying that up to isomorphism, $E$ is a collection of pairs $(b,v)$ where $b \in B$ and $v \in \chi^{-1}(h(b))$, where $\chi : E_{n,k} \to G_{n,k}$ is the tautological bundle over $G_{n,k}$. so in $h : B \to G_{n,k}$ the classifying map. In a sense we've "embedded" the vector bundle in Euclidean space, well, we've made the fibers subspaces of Euclidean space.

But again, you have the "for some $n$ sufficiently large" thing. And like with manifolds $n$ has an upper bound in terms of the dimension of $B$ (if $B$ is finite dimensional). Since it's sometimes awkward to carry-around these "for sufficiently large $n$" statements, you take the limit space $G_{\infty,k}$ and now your statement is far more clean, because any $k$-dimensional vector bundle over any space $B$ is the pull back of some map $B \to G_{\infty,k}$. The point is that $G_{\infty,k}$ is a universal space -- independent of $B$ or the vector bundle over $B$. The weak topology is exactly what allows us to ensure this happens.

show/hide this revision's text 1

There's been quite a bit of discussion of this thread in the meta thread. I'd like to take a stab at answering my interpretation of what the question is getting at. In a sense there's at least two things going on, and that's part of why I think there's been so much discussion.

To take a step back from Grassmannians, I'd like to mention why one might be interested in $\mathbb R^\infty$. The Whitney embedding theorem states that every continuous function $f : N \to \mathbb R^k$ can be approximated uniformly by a $C^\infty$-smooth embedding provided $k \geq 2n+1$ where the dimension of $N$ is $n$. Moreover, any two embeddings $N \to \mathbb R^k$ are isotopic provided $k \geq 2n + 2$. So if you wanted to, you could replace the class "$n$-manifolds up to diffeomorphism" with "isotopy classes of $n$-dimensional submanifolds of $\mathbb R^k$ provided $k \geq 2n+2$".

A key nice result about the weak topology on $\mathbb R^\infty$ is that any continuous function from a compact space to $\mathbb R^\infty$ has an image in some $\mathbb R^k$ for some $k$. So from the perspective of the Whitney embedding theorem above, "$n$-dimensional manifolds up to diffeomorphism" is precisely "$n$-dimensional submanifolds of $\mathbb R^\infty$ up to isotopy". The key thing here is the ambient space is now independent of the dimension of the manifold you're talking about. This is pretty much exactly what's going on with the Grassmannians.

Given a vector bundle $p : E \to B$ over a paracompact space $B$, there exists a classifying map for the bundle, meaning $p$ is isomorphic to the pull-back of the tautological bundle over $G_{n,k} \equiv G_k(\mathbb R^n)$. $n$ is just some sufficiently large integer. Although people don't state it this way, this theorem is basically the Whitney embedding theorem but for vector bundles. Because it's saying that up to isomorphism, $E$ is a collection of pairs $(b,v)$ where $b \in B$ and $v \in \chi^{-1}(h(b))$, where $\chi : E_{n,k} \to G_{n,k}$ is the tautological bundle over $G_{n,k}$. so in a sense we've "embedded" the vector bundle in Euclidean space, well, we've made the fibers subspaces of Euclidean space.

But again, you have the "for some $n$ sufficiently large" thing. And like with manifolds $n$ has an upper bound in terms of the dimension of $B$ (if $B$ is finite dimensional). Since it's sometimes awkward to carry-around these "for sufficiently large $n$" statements, you take the limit space $G_{\infty,k}$ and now your statement is far more clean, because any $k$-dimensional vector bundle over any space $B$ is the pull back of some map $B \to G_{\infty,k}$. The point is that $G_{\infty,k}$ is a universal space -- independent of $B$ or the vector bundle over $B$. The weak topology is exactly what allows us to ensure this happens.