I was watching an intro to vector bundles, and the good professor talks about a family of vector spaces that vary continuously. I've never come across this concept before, and am trying to think about how I'd formalize it. I'm following the example at 3:20, about the tangent line (in $\mathbb{R}^2$) to a circle.
If we fix a point $p$ on the circle, our tangent line can be considered as a one-dimensional vector space (say, parameterized by $t$) or as a 2D space (embedded in $\mathbb{R}^2$). In this way, we can consider the line as a function ($\mathbb{R} \rightarrow \mathbb{R}^2$) of $t$.
Then, letting $p$ vary, we can parameterize the circle by, say, $s$. Now our family of vector spaces becomes a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ of $(s, t)$.
Then we can define "continuity of vector spaces" to mean that for every $t$, the function $f(-, t)$ is continuous.
Is this approximately the right way to think about it?
What he meant by "vary continuously" is actually a sort of "compatible condition" we need to follow when we attach a family of vector spaces to points on a manifold to get a vector bundle.
To be more precise, for each point $p$ on the manifold, there exists an open neighborhood $U$ of $p$ such that the space $U \times \mathbb{R}^k$ and the union of all vector spaces that we attached to all points in $U$ look similar, i.e, there is a homeomorphism between them. Furthermore, the above homeomorphism restricting at any point in $U$ is also a linear isomorphism between vector spaces.
One way to formalize the circle example is the following:
Consider the unit circle, for each point $(\cos \theta, \sin \theta)$, attach a tangent line (vector space) $(\cos \theta -t\sin \theta, \sin \theta -t\cos \theta)$. So for each point $p=(\cos \theta, \sin \theta)$, there exists an $\epsilon$ such that we have the homeomorphism $$(\theta - \epsilon , \theta + \epsilon)\times \mathbb{R} \rightarrow \bigcup_{\alpha \in (\theta - \epsilon , \theta + \epsilon)} (\cos \alpha -t\sin \alpha, \sin \alpha -t\cos \alpha) $$ $$(\alpha_0,t_0) \mapsto (\cos \alpha_0 -t_0\sin \alpha_0, \sin \alpha_0 -t_0\cos \alpha_0) $$
At each point $\alpha_0 \in (\theta - \epsilon , \theta + \epsilon)$, the map $$(\alpha_0,t) \mapsto (\cos \alpha_0 -t\sin \alpha_0, \sin \alpha_0 -t\cos \alpha_0) $$ is a linear isomorphism between $\mathbb{R}$ and the tangent line attached at $\alpha_0$.