  A topological space is iso-dense if it has a dense set of isolated points. A
topological space is scattered if each of its non-empty subspaces has an
isolated point. In $\mathbf{ZF}$, in the absence of the axiom of choice, basic
properties of iso-dense spaces are investigated. A new permutation model is
constructed in which a discrete weakly Dedekind-finite space can have the
Cantor set as a remainder. A metrization theorem for a class of quasi-metric
spaces is deduced. The statement "every compact scattered metrizable space is
separable" and several other statements about metric iso-dense spaces are shown
to be equivalent to the countable axiom of choice for families of finite sets.
Results concerning the problem of whether it is provable in $\mathbf{ZF}$ that
every non-discrete compact metrizable space contains an infinite compact
scattered subspace are also included.
