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So the professor in class gave the following two theorems without proving ( or maybe I missed it).

Theorem 1. Let $M$,$N$ be two smooth manifolds. $L$ a submanifold of $N$. And $M$ is compact. Then the set of all maps $f:M\rightarrow N$ which are transverse to $L$ is open in $C^1$ topology.

Theorem 2. The set of $f:M\rightarrow N$ which are transverse to $L$ is dense in the $C^k$ topology for any $k$.

I've been looking for reference to find what the above two kinds of topologies are. In Hirsch's differential topology, the author introduced two kinds of topologies on the function space, namely the weak or "compact open $C^r$ topology" and the strong or Whitney topology. I wonder if the topology mentioned by my professor is any one of the two topologies here.

Thanks in advance.

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I guess that it is the CO topology. –  Sigur Feb 11 '13 at 22:04
    
@Sigur thanks, man –  henryforever14 Feb 12 '13 at 5:06

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