I was wondering about the cardinality of complete vector fields in relation to the cardinality of general vector fields.

Let $$\mathcal{A}=\{X:M\rightarrow TM| X \text{ is a vector field}\}$$ and $$\mathcal{B}=\{Y\in\mathcal{A}| Y \text{ is complete}\}$$

Can we say anything about $|\mathcal{A}|$ and $|\mathcal{B}|$? Obviously $|\mathcal{A}|\geq|\mathcal{B}|$, since $\mathcal{B}\subset\mathcal{A}$.

I'm trying to think as an analogy of the cardinality of continuous vs. all real functions of a real variable (in which case it differed as the former is just $|\mathbb{R}|$ whereas the latter, being the power set of $\mathbb{R}$ necessarily has a larger cardinality).

I've thought about it but the furthest I've gotten is realising that $|M|=|\mathbb{R}|$, since the charts are bijections between $M$ and $\mathbb{R}^n$ and we also have $|\mathbb{R}^n|=|\mathbb{R}|^n=|\mathbb{R}|$.


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