I'm looking for an inverse system $(X\_\alpha)\_{\alpha < \omega\_1}$ of vector spaces (**EDIT:** over a *finite* field) such that, for some $\lambda \geq 2$ with $\lambda < \lambda^{\omega\_1}$ (I believe the case where $\lambda = \kappa^\omega$ for some $\kappa \geq 2$ is particularly interesting), the following conditions hold:


1. The transition maps $X\_\alpha \to X\_\beta$, and the maps $\varprojlim\_{\alpha < \omega\_1} X\_\alpha \to X\_\beta$, are all surjective.
2. For $\alpha < \omega\_1$ a limit ordinal, the map $X\_\alpha \to \varprojlim\_{\beta < \alpha} X\_\beta$ is injective.
3. For $\alpha < \beta < \omega\_1$ and $\gamma < \omega\_1$ a limit ordinal, $\lambda = \dim X\_\alpha = \dim \ker(X\_\beta \to X\_\alpha) = \dim ((\varprojlim\_{\delta < \gamma} X\_\delta ) / X\_\gamma)$.
4. $\dim(\varprojlim\_{\alpha < \omega\_1} X\_\alpha) = \lambda^{\omega\_1}$.


Conditions (1) and (2) are basic structural conditions. Condition (3), in addition to stipulating that $X\_\alpha$ remain "small" below $\omega\_1$, also stipulates that the "rate of change" of $X\_\alpha$ be constant, at $\lambda$. The last part of (3) also rules out simply taking $X\_\alpha = V^\alpha$ for some $V$ of dimension $\lambda = \kappa^\omega$. The point of the cardinal $\lambda^{\omega\_1}$ in condition (4) is that it is the dimension of $\prod\_{\alpha < \omega\_1} X\_\alpha$, and thus the obvious upper bound subject to the other conditions.




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For comparison, consider an inverse system $(X\_n)\_{n \leq \omega}$ with $X\_\omega = \varprojlim\_{n < \omega} X\_n$ with surjective transition maps such that $1 < |X\_n| < \omega$ for all $n < \omega$. In this case, if $X\_\omega = \varprojlim\_{n <\omega} X\_n$, then there is a "jump" in cardinality at $\omega$, all the way up to the continuum. Part of the motivation of the above conditions is to look for similar "jumps" in cardinality at the bigger ordinal $\omega\_1$.


In fact, the full motivation of the above conditions is more specific, but in some sense it does boil down to asking for the growth to be "controlled" below $\omega\_1$, and suddenly "jump" at $\omega\_1$.



Example of an inverse system which suddenly "jumps" in size in a specific "controlled" way?
Here is my argument, which assumes $|2^\omega|<|2^{\omega\_1}|$ and uses an infinite base field. See Tim Campion's answer <https://mathoverflow.net/a/376790/164965> for the general case.


Take the base field to be $\mathbb Q,$ set $\lambda=2^\omega=\mathfrak c,$ and $X\_{\alpha}=\ell^\infty(\alpha)$: the bounded functions $\alpha\to\mathbb R.$
(If you prefer, you could use simple functions instead of all bounded functions.) The transition map $X\_\alpha\to X\_\beta$ is the restriction to $\beta.$


For each $\alpha>0,$ the space $X\_\alpha$ has $\mathbb Q$-dimension $\mathfrak c.$ If $\dim X\_0$ matters, take the example $X'\_\alpha=X\_{1+\alpha}$ instead.


For $\alpha<\beta<\omega\_1$ the transition map has a kernel of $\mathbb R$-dimension at least one, so $\mathbb Q$-dimension $\mathfrak c.$


The limit $\varprojlim\_{\delta<\gamma} X\_\delta$ at countable limit ordinals $\gamma$ is the set of functions $\gamma\to\mathbb R$ that are bounded when restricted to $\delta$ for any $\delta<\gamma.$ This set contains $X\_\gamma$ as a proper $\mathbb R$-subspace, so the $\mathbb Q$-dimension of $(\varprojlim\_{\delta<\gamma} X\_\delta)/X\_\gamma$ is $\mathfrak c.$


The big limit $\varprojlim\_{\alpha < \omega\_1} X\_\alpha$ is just $\ell^\infty(\omega\_1)$ because any unbounded function on $\omega\_1$ would be unbounded on some $\alpha<\omega\_1.$ The dimension of this space is its cardinality, $2^{\omega\_1}.$


