  In a previous paper ("A characterization of topologically completely positive
entropy for shifts of finite type"), the author gave a characterization for
when a $\mathbb{Z}^d$-shift of finite type (SFT) has no nontrivial subshift
factors with zero entropy, a property which we here call zero-dimensional
topologically completely positive entropy (ZTCPE). In this work, we study the
difference between this notion and the more classical topologically completely
positive entropy (TCPE) of Blanchard. We show that there are one-dimensional
subshifts and two-dimensional SFTs which have ZTCPE but not TCPE. In addition,
we show that strengthening the hypotheses of the main result of the
aforementioned paper yields a sufficient condition for a $\mathbb{Z}^d$-SFT to
have TCPE.
