In this paper, we prove the Soul Conjecture in Alexandrov geometry in dimension $4$, i.e. if $X$ is a complete non-compact $4$-dimensional Alexandrov space of non-negative curvature and positive curvature around one point, then a soul of $X$ is a point.
Given two $n_i$-dimensional Alexandrov spaces $X_i$ of curvature $ge 1$, the join of $X_1$ and $X_2$ is an $(n_1+n_2+1)$-dimensional Alexandrov space $X$ of curvature $ge 1$, which contains $X_i$ as convex subsets such that their points are $frac pi2
$ apart. If a group acts isometrically on a join that preserves $X_i$, then the orbit space is called quotient of join. We show that an $n$-dimensional Alexandrov space $X$ with curvature $ge 1$ is isometric to a finite quotient of join, if $X$ contains two compact convex subsets $X_i$ without boundary such that $X_1$ and $X_2$ are at least $frac pi2$ apart and $dim(X_1)+dim(X_2)=n-1$.
