$\mathbf{The \ Problem \ is}:$ Let $X$ is a locally compact, separable metric space. Let $G$ be an abelian group. Now I came across two definitions of cohomological dimension of $X.$ One is the usual definition $\textbf{(P)}$ : $X$ is said to have cohomological dimension $n$ if $n$ is the largest integer such that there exists a closed set $A\subset X$ with the relative Čech cohomology of the pair $(X,A)$ $\check{H}^n(X,A;G)\neq 0.$ But for locally compact spaces, I got another definition $\textbf{(Q)}$ : $X$ is said to have cohomological dimension $n$ if $n$ is the largest integer such that there exists a locally compact $A\subset X$ with $H^n_c(A;G)\neq 0.$ For the $2$nd definition, see page $5$ of V. Kuz’minov's paper “Homological Dimension Theory”. Now my query is why $\textbf{(P)}$ and $\textbf{(Q)}$ are equivalent for locally compact spaces?

$\mathbf{My \ Approach}:$ Suppose $\textbf{(Q)}$ holds to be true. Let $X^{+}:=X\cup x^{+}$ be an one-point compactification of $X.$ Then we have a natural isomorphism between $H^n_c(X^{+}\setminus A;G)$ with that of $\check{H}^n(X^{+},A;G).$ Now by $\textbf{(P)}$, we can get an open set $U$ with $\bar{U}$ compact such that $H^n_c(U;G)\neq 0.$ Then the above isomorphism gives us $\operatorname{dim}_GX\geq n$ with respect to $\textbf{(Q)}$. Now it can be shown(also done in Theorem $1$ of page $5$ in Kuzminov's paper "Homological Dimension Theory") that $\textbf{(P)}$ is equivalent to the fact that $j^n_{A,B} : H^n_c(A;G)\to H^n_c(B;G)$ is a surjection for every pair of compact sets $B\subset A$ in $X.$ Then by the long exact sequence $$\to H^n_c(A;G)\to H^n_c(X^{+};G)\to H^{n+1}_c(X^{+}\setminus A;G)\to$$ we get $H^{n+1}_c(X^{+}\setminus A;G)=0$ implying $\check{H}^n(X^{+},A;G)=0$ for each closed $A\subset X$ which implies $\operatorname{dim}_GX\leq n$ with respect to $\textbf{(Q)}$. Hence $\textbf{(P)}$ $\implies$ $\textbf{(Q)}$. Now I am facing problem in showing $\textbf{(Q)}$ implies $\textbf{(P)}$. Can we say that the isomorphism between $H^n_c(X\setminus A;G)$ and $\check{H}^n(X,A;G)$ holds for locally compact spaces $X$ and closed subset $A$ of $X ?$ Then we could say that there exists an open $U\subset X$ with $\bar{U}$ compact(by passing to the limit as $X$ is locally compact) such that $H^n_c(U;G)\neq 0$ whcih implies $\operatorname{dim}_GX\geq n$ with respect to $\textbf{(P)}$ and as $\check{H}^n(X,C;G)=0$ for all closed $C\subset X,$ hence $\operatorname{dim}_GX\leq n$ with respect to $\textbf{(P)}$ giving us $\textbf{(Q)}$ $\implies$ $\textbf{(P)}$. But I can't say that those two cohomology groups are isomorphic when $X$ is locally compact ?