For a separable metric space X, we consider possibilities for the sequence where . In Section 1, a general method for producing examples is given which can be used to realize many of the possible sequences. For example, there is such that , , for n >1, such that , and Z such that S(Z) = 4, 4, 6, 6, 7, 8, 9,.... In Section 2, a subset X of is shown to exist which satisfies and .
We give an example of an extremally disconnected compact Hausdorff space with an open continuous selfmap such that the fixed point set is nonvoid and nowhere dense, respṫhat there is exactly one nonisolated fixed point.
We prove that the Niemytzki plane is -metrizable and we try to explain the differences between the concepts of a stratifiable space and a -metrizable space. Also, we give a characterisation of -metrizable spaces which is modelled on the version described by Chigogidze.
We construct from ⋄ a T₂ example of a hereditarily Lindelöf space X that is not a D-space but is the union of two subspaces both of which are D-spaces. This answers a question of Arhangel'skii.
We define -directedness, investigate various properties to determine whether they have this property or not, and use our results to obtain easier proofs of theorems due to Laurence and Alster concerning the existence of a Michael space, i.eȧ Lindelöf space whose product with the irrationals is not Lindelöf.