Reference request: Topology on the space of smooth compact submanifolds In Allen Hatchers short exposition of the Madsen-Weiss Theorem he defines the topology on the space $\mathcal{C}^n$ of smooth oriented properly embedded $d$-dimensional submanifolds of $\mathbb{R}^n$ on page 6 with the help of the 
"standard $C^\infty$ topology on the space of d-dimensional smooth
compact submanifolds of B that are properly embedded (meaning that the intersection of the submanifold with $\partial B$ is the boundary of the submanifold, and this is a
transverse intersection)",
where $B$ is a closed ball centered in the origin.
I am searching for a reference or an explicit definition of the "standard topology" Hatcher is referring to.
 A: It's in Hirsch's "Differential Topology" textbook.  Specifically, given a compact manifold $M$, the weak $C^k$-topology on the set of $C^k$-smooth embeddings $Emb(M,\mathbb R^n)$ is the one that requires uniform closeness of not only the maps themselves, but all derivatives up to order $k$.   For $C^\infty$ mapping spaces you demand uniform agreement up to some arbitrary order $k$ -- it is not a normable space anymore so you think of the topology as being induced by a countable sequence of (semi)norms, one for each $k$.
That's the space of embeddings.  So the space of submanifolds of $\mathbb R^n$ diffeomorphic to $M$ is the space 
$$Emb(M,\mathbb R^n) / Diff(M) $$
where $Diff(M)$ is given the same "weak" topology as in Hirsch. $Diff(M)$ is the space of $C^k$-diffeomorphisms of $M$.  Notice two embeddings $M \to \mathbb R^n$ have the same image if and only if they differ by a diffeomorphism of $M$.
Then the space of all $m$-dimensional manifolds in $\mathbb R^n$ is the disjoint union:
$$\sqcup_M Emb(M, \mathbb R^n) / Diff(M)$$
where you take the disjoint union over all diffeomorphism types of $m$-dimensional manifolds $M$. 
It's a jazzed-up version of the Whitney embedding theorem that states $Emb(M,\mathbb R^n)$ is highly-connected for $n$ much larger than $m$. 
