This question should be fairly elementary. It is related to this question, and I’d just like to check I’m not missing anything.

Let $\{M_n\}_{n\ge 0}$ be an inverse system of smooth manifolds, resp smooth complex analytic spaces, with transition maps $f_{t,s} : M_t\to M_s$, $t\ge s$, that are local diffeomorphisms, resp local analytic isomorphisms.

Let $M$ be the topological inverse limit.

For every $x\in M$, does there exist an open neighborhood $U\subset M$ of $x$, such that $U$ is homeomorphic to some smooth manifold, resp smooth complex analytic space, that is diffeomorphic, resp locally analytically isomorphic, to an open subset of $M_n$ for some $n$?

This is certainly true for each of the finite layers, and should be true in the limit. In particular, $M$ would be locally compact Hausdorff and second countable, and one may define a notion of “smooth” partition of unity, and give a sufficient answer to the linked question.

This question should be fairly elementary. I’d just like to check I’m not missing anything.

Let $\{M_n\}_{n\ge 0}$ be an inverse system of smooth manifolds with transition maps $f_{t,s} : M_t\to M_s$, $t\ge s$, that are local diffeomorphisms.

Let $M$ be the topological inverse limit.

For every $x\in M$, does there exist an open neighborhood $U\subset M$ of $x$, such that $U$ is homeomorphic to some smooth manifold that is diffeomorphic to an open subset of $M_n$ for some $n$?

This question should be fairly elementary. It is related to this question, and I’d just like to check I’m not missing anything.

Let $\{M_n\}_{n\ge 0}$ be an inverse system of smooth manifolds, resp smooth complex analytic spaces, with transition maps $f_{t,s} : M_t\to M_s$, $t\ge s$, that are local diffeomorphisms, resp local analytic isomorphisms.

Let $M$ be the topological inverse limit.

For every $x\in M$, does there exist an open neighborhood $U\subset M$ of $x$, such that $U$ is homeomorphic to some smooth manifold, resp smooth complex analytic space, that is diffeomorphic, resp locally analytically isomorphic, to an open subset of $M_n$ for some $n$?

This is certainly true for each of the finite layers, and should be true in the limit. In particular, $M$ would be locally compact Hausdorff and second countable, and one may define a notion of “smooth” partition of unity, and give a sufficient answer to the linked question.

Remark: Note that the sought for condition does not imply $M$ itself is a manifold, resp complex analytic space. We would need to check cocycle conditions on a suitable cover, and those are satisfied only in the limit. Since I am not assuming any compactness on the $M_n$, let alone $M$, such cover will usually be infinite and an infinite number of cocycle conditions should be satisfied. We cannot descend them all to some common finite layer, then.

This question should be fairly elementary. It is related to this question, and I’d just like to check I’m not missing anything.

Let $\{M_n\}_{n\ge 0}$ be an inverse system of smooth manifolds, resp smooth complex analytic spaces, with transition maps $f_{t,s} : M_t\to M_s$, $t\ge s$, that are local diffeomorphisms, resp local analytic isomorphisms.

Let $M$ be the topological inverse limit.

For every $x\in M$, does there exist an open neighborhood $U\subset M$ of $x$, such that $U$ is homeomorphic to some smooth manifold, resp smooth complex analytic space, that is diffeomorphic, resp locally analytically isomorphic, to an open subset of $M_n$ for some $n$?

This is certainly true for each of the finite layers, and should be true in the limit. In particular, $M$ would be locally compact Hausdorff and second countable, and one may define a notion of “smooth” partition of unity, and give a sufficient answer to the linked question.

This question should be fairly elementary. It is related to this question, and I’d just like to check I’m not missing anything.

Let $\{M_n\}_{n\ge 0}$ be an inverse system of smooth manifolds, resp smooth complex analytic spaces, with transition maps $f_{t,s} : M_t\to M_s$, $t\ge s$, that are local diffeomorphisms, resp local analytic isomorphisms.

Let $M$ be the topological inverse limit.

For every $x\in M$, does there exist an open neighborhood $U\subset M$ of $x$, such that $U$ is homeomorphic to some smooth manifold, resp smooth complex analytic space, that is diffeomorphic, resp locally analytically isomorphic, to an open subset of $M_n$ for some $n$?

This is certainly true for each of the finite layers, and should be true in the limit. In particular, $M$ would be locally compact Hausdorff and second countable, and one may define a notion of “smooth” partition of unity, and give a sufficient answer to the linked question.

Remark: Note that the sought for condition does not imply $M$ itself is a manifold, resp complex analytic space. We would need to check cocycle conditions on a suitable cover, and those are satisfied only in the limit. Since I am not assuming any compactness on the $M_n$, let alone $M$, such cover will usually be infinite and an infinite number of cocycle conditions should be satisfied. We cannot descend them all to some common finite layer, then.

This question should be fairly elementary. It is related to this question, and I’d just like to check I’m not missing anything.

Let $\{M_n\}_{n\ge 0}$ be an inverse system of smooth manifolds, resp smooth complex analytic spaces, with transition maps $f_{t,s} : M_t\to M_s$, $t\ge s$, that are local diffeomorphisms, resp local analytic isomorphisms.

Let $M$ be the topological inverse limit.

For every $x\in M$, does there exist an open neighborhood $U\subset M$ of $x$, such that $U$ is homeomorphic to some smooth manifold, resp smooth complex analytic space, that is diffeomorphic, resp locally analytically isomorphic, to an open subset of $M_n$ for some $n$?

This is certainly true for each of the finite layers, and should be true in the limit. In particular, $M$ would be locally compact Hausdorff and second countable, and one may define a notion of “smooth” partition of unity, and give a sufficient answer to the linked question.

Remark: Note that the sought for condition does not imply $M$ itself is a manifold, resp complex analytic space. We would need to check cocycle conditions on a suitable cover, and those are satisfied only in the limit. Since I am not assuming any compactness on the $M_n$, let alone $M$, such cover will usually be infinite and an infinite number of cocycle conditions should be satisfied. We cannot descend them all to some common finite layer, then.