It is shown that if $X$ is a Stein space and $S$ is a closed set in $X$ with $H_{2\dim X-1}(S)=0$, then $H(X)\in (DN)$ if and only if $H(X\mathrm{\setminus }S)\in (DN)$. Moreover it is also shown that the property $(DN)$ is invariant under holomorphic surjections between Stein spaces with connected fibres having the property $(DN)$.