Let $\left( X, x_0\right) $ be a pointed topological space. I don not require that $X$ is Hausdorff. Suppose $X$ is contractible, i.e. that there is a homotopy $F:X \times [0,1] \to X$ such that $F(0)= Id_X$ and $F(1) = \{x_0\}$. Let $\widehat{\mathbb{Z}}$ be a locally constant $\mathbb{Z}$-sheaf. I think that $H^p\left(X , \widehat{ \mathbb{Z}} \right) = \{0\}$ for all $p > 0$ where $H^p$ are sheaf cohomology explained in Hartshorne's book. Is it true? If it is true, how this fact can be proven.There is a theorem in Bredon's book (page 80). \begin{theorem} Any two properly homotopic maps (with respect to $\Phi$ and $\Psi$) of a space $X$ into a space $Y$ induce identical homomorphisms $H^_\Psi(Y;G)\to H^_\Phi(X;G)$, where $G$ is any constant coefficient group.
\end{theorem} Note the special cases: (a) $\Phi = cld = \Psi$. In this case, "properly homotopic" is the same as "homotopic." (b) $X, Y$ locally compact Hausdorff, $\Phi = c = \Psi$. Is the case (a) is relevant to my situation?