We show that every $T_{0}$ space $X$ has some $T_{0}$ "special" one-point connectification $X_{\infty}$, unique up to a homeomorphism, such that $X$ is a closed subspace of $X_{\infty}$ and a closed subset of $X$ is precisely a closed proper subset of $X_{\infty}$; moreover, having such a one-point connectification characterizes $T_{0}$ spaces. As an application, it is also shown that our one-point connectification of every given topological $n$-manifold is a space more general than but "close to" a topological $n$-manifold with boundary.