Topology of a manifold A manifold $M$ is a locally euclidian topological space (every point has a neighborhood homeomorphic to an open subset of $\mathbb{R}^n$).  We assume, in addition, $M$ Haussdorf and second countable.
My question is: are the following properties equivalent?
$M$ is $\sigma$-compact (or countable at infinity).
$M$ has a countable atlas.
$M$ is metrizable and separable.
$M$ is paracompact.
$M$ admits a partition of unity subordinate to any open cover.
Are the following results 'easy' to prove:


*

*Any atlas of class $C^k$ ($k\geq1$) is $C^k$-compatible with a $C^\infty$ atlas.

*Any manifold has a finite atlas (using the theory of dimension?).
Thank you for your help.
 A: About the topology of manifolds we have 
paracompactness $\iff$ partitions of unity $\iff$ every component has a countable basis $\iff$ ...
Not hard to prove.
About differentiable structures, it is more convenient to talk about diffeomorphisms.
Here are the facts (classic results from the $50$'s) -- (assume paracompact to be safe)
Let $M$ a $C^k$ manifold ($k\ge 1$). Then there exists a $C^{\omega}$ manifold ( real analytic) $\tilde M$ and a $C^k$ diffeomorphism $M  \simeq \tilde M$ ( this is equivalent to: there exists a $C^{\omega}$ atlas on $M$ which is $C^k$-compatible with the given $C^k$ atlas.
Let $1\le k 
\le l$  $M$, $M'$ two $C^l$ manifolds that are isomorphic as $C^k$ manifolds. Then they are isomorphic as $C^l$ manifolds. From the previous statement only the case $k=1$, $l=\omega$ has to be considered. 
Whitney  showed  that every differentiable manifold is a submanifold of some $\mathbb{R}^N$ with image real analytic. This implies the existence of a real analytic structures.
John Nash ( of a beautiful mind) proved that every Riemannian manifold  can be imbedded in some $\mathbb{R}^n$ with image a real analytic submanifold. 
The previous results do not work for $k=0$: 
There exists a  topological ($C^0$ ) manifold that is not homeomorphic to a ($C^1$) differentiable manifold ( Kervaire).
There exist non-diffeomorphic differetiable manifolds that are homeomorphic (Milnor).
See also exotic $\mathbb{R}^4$. 
