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.



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> **Q1.** Can we conclude that for every $a\in J'$ there exists a $b\in J$ such that $H\_a\subseteq K\_b$?
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> 
> **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?
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The [answer](https://mathoverflow.net/a/415828) by KP Hart answers Q1 in the negative.



Approximations by compact sub-spaces
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.


