entropy is in some sense completely positive, that is, the Kolmogorov-Sinai entropy of every nontrivial partition is positive, equivalently, the measure-theoretic entropy of every nontrivial measure-preserving factor is positive. It is natural to seek for an analog of this notion in topological dynamics. It turns out that the conditions characterizing $K$-systems in ergodic theory are no longer equivalent when translated to the topological setting. This problem was studied by Blanchard [@B92] who defined *completely positive entropy* and *uniform positive entropy*. A dynamical system $(X,T)$ has *completely positive entropy* if all nontrivial topological factors of this system have positive topological entropy, and $(X,T)$ has *uniform positive entropy* if $T$ has positive topological entropy with respect to every open cover of $X$ by two sets none of which is dense in $X$. Blanchard proved that uniform positive entropy implies completely positive entropy (this was also proved earlier by Bowen, see Proposition 4.2 in [@Bowen71]), but the converse implication is not true. Moreover, completely positive entropy does not imply any mixing property. Huang and Ye [@HY06] introduced the notion of a *topological $K$-system*. Following them we say that $(X,T)$ is a *topological $K$-system* if every finite cover of $X$ by nondense and open 