CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
Given a continuous function between two connected closed oriented manifolds of the same dimension, its degree is a measure for how often the function “wraps its domain around its codomain”.
For $X$ is a connected closed oriented manifold of dimension $n$, its top homology group $H_n(X) = H_n(X; \mathbb{Z})$ is isomorphic to $\mathbb{Z}$, where the generator $1 \in \mathbb{Z}$ is identified with the orientation class $[\omega_X]$ of $X$, the fundamental class of $X$.
Given a continuous map $f \colon X \to Y$ between two such manifolds, the homomorphism $f_\ast = H_n(f) \colon H_n(X) \to H_n(Y)$ is therefore specified by the integer $n$ such that $f_\ast [\omega_X] = n [\omega_Y]$. This integer is called the degree of $f$.
We suppose throughout that $X$ and $Y$ are connected closed oriented manifolds of the same dimension $n$. The degree of a continuous function $g \colon X \to Y$ is frequently computed according to the following considerations:
The space of continuous functions $g \colon X \to Y$ has a dense subspace consisting of smooth functions $f \colon X \to Y$, and in particular every continuous function $g$ is homotopic to a smooth function $f$. It therefore suffices to compute the degree of $f$.
By Sard's theorem?, the set of singular values? of a smooth function $f$ has measure zero (using for example the orientation on $Y$ to define a volume form and hence a measure). Accordingly, we may choose a regular value? $y \in Y$.
The inverse image $f^{-1}(y)$ is a compact $0$-dimensional manifold, hence consists of finitely many (possibly zero) points $x_1, \ldots, x_r \in X$. Since these are regular points, $f$ restricts to a diffeomorphism
where $U_i$ is a small neighborhood of $x_i$ and $V$ is a small neighborhood of $y$. The diffeomorphism $f_i$ either preserves or reverses the orientation of $U_i$, i.e., the sign of the determinant as a mapping between differential n-forms
is either $+1$ or $-1$.
By a straightforward application of the excision axiom? in homology, it follows that the degree of $f$ is the sum of these signs:
and this quantity is independent of the choice of regular value $y$.