Continuous singular simplex and real singular chain group

For a topological space $X$ and integer $k\ge 0$, let $S_k(X)$ be the set of continuous maps ${\mathrm{\Delta }}^k\to X$ from the standard simplex, as in Singular simplices and singular chain groups with coefficients. Define $C_k(X;\mathbb{R})={\mathbb{R}}^{(S_k(X))}=\{a:S_k(X)\to \mathbb{R}:\{\sigma :a(\sigma )\mathrm{\ne }0\}\text{ is finite}\}\mathrm{.}$ Addition and scalar multiplication are pointwise. Write $[\sigma ]$ for the function equal to $1$ at $\sigma$ and $0$ elsewhere. Every chain has the unique expression ${\sum }_{\sigma }{a}_{\sigma }[\sigma ]$ over its finite support. Set $C_k(X;\mathbb{R})=0$ for $k<0$.

This is the real specialization of the cited tensor convention: the balanced bilinear map $(\sum n_{\sigma }[\sigma ],r)\mapsto \sum r{n}_{\sigma }[\sigma ]$ induces $C_k(X;\mathbb{Z}){\otimes }_{\mathbb{Z}}\mathbb{R}⟶{\mathbb{R}}^{(S_k(X))}\mathrm{.}$ Its inverse sends $\sum a_{\sigma }[\sigma ]$ to $\sum [\sigma ]\otimes a_{\sigma }$. Additivity follows by collecting the finite union of supports; the tensor relations show the composites fix every $[\sigma ]\otimes r$ and every $a[\sigma ]$. Thus these are inverse real-linear maps. No basis selection or axiom of choice is used: the simplex basis is part of the definition.

If $X=\mathrm{\varnothing }$ there are no simplices and the chain space is zero in every degree. If $X$ is one point, there is one simplex in every nonnegative degree and its chain space is $\mathbb{R}$. Constant and degenerate simplices are retained; this is the unnormalized convention.