Real singular cohomology

For the complex in Real singular cochain complex, define the real vector spaces $Z^k(X;\mathbb{R})=\ker {\delta }^k,B^k(X;\mathbb{R})=\mathrm{im}{\delta }^{k-1},H_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}^k(X;\mathbb{R})=Z^k(X;\mathbb{R})\mathrm{/}{B}^k(X;\mathbb{R})\mathrm{.}$ The square-zero identity puts $B^k$ inside $Z^k$, so the quotient is well-defined. This is the vector-space instance of Cohomology object of a cochain complex. A cocycle has $\delta \phi =0$, and a coboundary is $\delta \psi$. Two cocycles determine the same class if and only if their difference is a coboundary. Addition and scalar multiplication are induced by those of cocycles; changing representatives adds a coboundary since $B^k$ is a vector subspace.

Negative cohomology groups are zero. At degree zero there are no incoming coboundaries: $H^0$ is the space of point functions taking equal values on the endpoints of every path. For the empty space every group is zero; for a point the alternating differential calculated in the preceding definition gives $H^0=\mathbb{R}$ and $H^k=0$ for $k\mathrm{\ne }0$.

This definition uses the cochain quotient, without choosing representatives or identifying it with a dual homology space. It requires no AC.