  This article continues the study of computable elementary topology started by
the author and T. Grubba in 2009 and extends the author's 2010 study of axioms
of computable separation. Several computable T3- and Tychonoff separation
axioms are introduced and their logical relation is investigated. A number of
implications between these axioms are proved and several implications are
excluded by counter examples, however, many questions have not yet been
answered. Known results on computable metrization of T3-spaces from M.
Schr/"oder (1998) and T. Grubba, M. Schr/"oder and the author (2007) are proved
under uniform assumptions and with partly simpler proofs, in particular, the
theorem that every computably regular computable topological space with
non-empty base elements can be embedded into a computable metric space. Most of
the computable separation axioms remain true for finite products of spaces.
