The reduced degree of a map into the 0-sphere

Write $S^0=\{-1,+1\}=\mathrm{\partial }[-1,1]$ for the $0$-sphere (Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$) and orient it by the boundary orientation of the interval $[-1,1]$ (Induced boundary orientation): the point $+1$ carries the sign $+1$ and the point $-1$ the sign $-1$, since an orientation of a $0$-manifold is a sign at each point (Oriented smooth manifolds and oriented charts). Let $(P,s)$ be a finite oriented $0$-manifold, that is, a finite set $P$ together with a sign $s(x)\in \{+1,-1\}$ for every $x\in P$ (Oriented smooth manifolds and oriented charts); it is balanced when ${\sum }_{x\in P}s(x)=0.$ For a balanced $(P,s)$ and a map $f:P\to S^0$ — every map is continuous because $P$ is discrete and $S^0$ is discrete — define the reduced degree $\mathrm{deg}(f):=\frac{1}{2}{\sum }_{x\in P}s(x)f(x)\in \mathbb{Z},$ the signed count of the values divided by two, where the values $f(x)\in \{-1,+1\}$ are read in $\mathbb{R}$. This is well defined: since $f$ takes only the two values $\pm 1$, balance gives ${\sum }_{f(x)=+1}s(x)=-{\sum }_{f(x)=-1}s(x)$, hence ${\sum }_{x}s(x)f(x)=2{\sum }_{f(x)=+1}s(x)$ is even, and replacing $s$ by $-s$ changes the sign of $\mathrm{deg}(f)$ exactly as changing an orientation changes the ordinary degree (Degree of a map between oriented closed manifolds).

The case $P=S^0$ with the standard orientation above is balanced, and then $\mathrm{deg}(f)=\frac{f(+1)-f(-1)}{2}\in \{-1,0,+1\}\mathrm{.}$ This is the induced map on the reduced group ${\tilde{H}}_0(S^0;\mathbb{Z})\cong \mathbb{Z}$ generated by the difference of the two point classes (Reduced homology theory and augmentation): the identity has reduced degree $+1$, the antipodal map $-1$, and the two constant maps $0$. It is the scalar replacement for the fundamental-class degree of Degree of a map between oriented closed manifolds, whose scalar definition is stated only for nonempty connected closed oriented manifolds and therefore does not apply to a source as disconnected as $S^0$.

The motivating balanced sources are boundaries: if $W$ is a compact oriented smooth $1$-manifold and $\mathrm{\partial }W$ carries the induced boundary orientation, then ${\sum }_{p\in \mathrm{\partial }W}\epsilon (p)=0$ (Oriented boundary counts of a compact oriented 1-manifold cancel), so $\mathrm{\partial }W$ is balanced and every map $\mathrm{\partial }W\to S^0$ has a reduced degree. No choice principle is used in the definition; the cited boundary-count lemma assumes ${\mathrm{A}\mathrm{C}}_{\omega }$.