There are two different ways to define a degree of map.
Let $M$, $N$ be smooth, and $M$ is compact, $N$ is connected. If $f\in C(M,N)$, we define $\deg f$ by smooth approximation. [Milnor/Topology from a Differential Viewpoint]
Let $M$, $N$ be finite pseudomanifold, and $f\in C(M,N)$, $\deg f$ can be defined by simplical approximation. [Spanier/Algebra Topology/Page 207]
In the second, both spaces are compact, so it is smaller than the first one from this point. So my question is
Can we remove the finiteness of $N$ in the second definition?
Any advice is helpful. Thank you.