Let $\Delta_{\ast}(X,A)$ (resp. $\Delta_{\ast}^c(X,A)$) be the chain complex of locally finite (resp. finite) singular chains of $X$ modulo those chains in $A$.
How to show that the homomorphism of generated sheaves induced by the natural homomorphism $$\Delta_{\ast}^c(X,X\backslash U)\longrightarrow\Delta_{\ast}(X,X\backslash U)$$ of presheaves is an isomorphism ? Is that because $\Delta_{\ast}^c(X,X\backslash\{x\})\simeq\Delta_{\ast}(X,X\backslash\{x\})$ ?
Then, let this generated sheaf be denoted by $\Delta_{\ast}$.How to show that the presheaf $U\mapsto\Delta_{\ast}(X,X\backslash U)$ (which generates $\Delta_{\ast}$) is a monopresheaf and that it is conjunctive for coverings of $X$?
Lastly, I'd like to show that $$\theta :\Delta_{\ast}(X)\rightarrow\Gamma(\Delta_{\ast})$$ is a isomorphism when $X$ is paracompact...