TITLE: Is limit of null-homotopic maps null-homotopic?
QUESTION [10 upvotes]: The question is motivated by my failed comment to this one.
Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds).
Let $\varphi_{n}:M\to N$ be null-homotopic and convergent to $\varphi:M\to N$ in the compact open topology.

Does it follow that $\varphi$ is null-homotopic?

Note that homotopy between maps is a path in $C(M,N)$ (for nice $M$, $N$), and so what my question asks is whether the path component of a constant map is closed in $C(M,N)$.
I am waaay out of my depth here, but perhaps there is a continuous or positive lower semi-continuous functional on $C(M,N)$ akin to the topological degree such that null-homotopic maps would be the zero-set of that functional?

REPLY [2 votes]: Please see the answer to Annie's question.
Non-density of continuous functions to interior in set of all continuous functions