  The aim of this paper is to study the topological properties of algebraic
sets with zero divisors. We impose a subbasic topology on the set of proper
ideals of a $k$-algebra and this new ``$k$-space'' becomes a generalization of
the corresponding Zariski space. We prove that a $k$-space is
$T_{\scriptscriptstyle 0}$, quasi-compact, spectral, and connected. Moreover,
we study continuous maps between such $k$-spaces. We conclude with a question
about construction of a sheaf of $k$-spaces similar to affine schemes.
