Approximations by compact sub-spaces Suppose $X$ is a Hausdorff (I'm happy to also assume "non compact") topological space that can be written as the topological direct limit
$$\varinjlim_{a\in J} K_a$$
for $J$ a directed set and $K_a$ compact Hausdorff subspaces of $X$, where all bonding maps are continuous non-surjective immersions (I'm happy to assume, additionally, that they are not surjective).
Suppose that there is another system of compact Hausdorff subspaces of $X$ with bonding maps as in the previous system,
$$\{H_a, a\in J'\}$$
for a directed set $J'$ with $J\subseteq J'$. Suppose that for every $a\in J$ we have
$$K_a\subseteq H_a.$$
Suppose that we still have
$$X = \varinjlim_{a\in J'}H_a$$
with the direct limit topology on the right side and $X$'s own topology on the left side.

Q1. Can we conclude that for every $a\in J'$ there exists a $b\in J$ such that $H_a\subseteq K_b$?


Q2. Can we conclude as in Q1 if in addition all the $K_a$'s and $H_a$'s are compact differentiable manifolds with differentiable immersions as bonding maps?

The answer by KP Hart answers Q1 in the negative.
 A: Q1. Consider the space of rationals $\mathbb{Q}$.
It is the direct limit of the family of finite unions of convergent sequences (including their limits), ordered by inclusion, as a set is closed iff its intersection with every convergent sequence is closed.
Now choose for every convergent sequence $\mathbf{q}=\langle q_n:n\in\mathbb{N}\rangle$ a set $S_\mathbf{q}$ of convergent sequences $\{\mathbf{r}_n:n\in\mathbb{N}\}$, where $\mathbf{r}_n$ converges to $q_n$, and has diameter not larger than $1/n$.
If $K$ is a finite family of convergent sequences then let $H_K=\bigcup_{\mathbf{q}\in K}S_\mathbf{q}$.
Then $K\subseteq H_K$, and $H_K$ is compact; and $\mathbb{Q}$ is also the direct limit of the system of $H_K$s. But there are no $L$ and $K$ such that $H_K\subseteq L$ ($L$ has finitely many accumulation points, $H_K$ has infinitely many).
Addendum: this construction works in every metric space without isolated points.
There a set is closed iff its intersection with every convergent sequence is closed, and every point is the limit of a non-trivial convergent sequence of arbitrarily small diameter.
