My background is primarily in algebra and topology/geometry with my primary interest lying in algebraic geometry. I am learning about locally free sheaves in the context of schemes, and they always seem to motivate it with the example of vector bundles. I was hoping I would be able to get a nice sheafy description of vector bundles on a manifold (viewed as a locally ringed space), but it doesn't seem to be working out as I expected. I think part of e problem is that my analysis background is considerably weaker.
This is what I have been thinking so far. Let $M$ be a real topological $n$-manifold (I'll worry about smoothness later once I have ironed out the details). I'll write it as $(M, \mathcal{O}_{M})$ where $\mathcal{O}_{M}$ is the sheaf of continuous real-valued functions. That is, $(M, \mathcal{O}_{M})$ is a locally ringed space which is locally isomorphic to $(\mathbb{R}^{n}, \mathcal{C}^{0})$. Let $\mathcal{L}$ be a locally free sheaf of $\mathcal{O}_{M}$-modules of rank $1$.
Now define the sheaf space, $$ \text{sp}(\mathcal{L}) = \bigcup_{p \in M} \mathcal{L}_{p}, $$ with a projection $\pi: \text{sp}(\mathcal{L}) \longrightarrow M$ defined by $\pi(s) = p$ if $s \in \mathcal{L}_{p}$. Then give $\text{sp}(\mathcal{L})$ the coarsest topology making $\pi$ continuous.
I would like to make this into a vector bundle of rank $1$ (if this is even possible?). But in order to do that, I'd need $\pi^{-1}(p) = \mathbb{R}$. That is, $$ \pi^{-1}(p) = \mathcal{L}_{p} \simeq \mathcal{C}^{0}_{p} \stackrel{?}{\simeq} \mathbb{R}. $$ My problem is with the isomorphism labelled with '?'. This is where my analysis knowledge is failing me, but the idea that the ring of germs of continuous real-valued functions at a point is $\mathbb{R}$ seems absurd to me, although certainly the residue field by its maximal ideal would be.
Is this actually the correct approach to see a correspondence between vector bundles and locally free sheaves, or am I totally on the wrong track?