In this paper it is studied the role of the axiom of choice in some theorems in which the concepts of first and second countability are used. Results such as the following are established: (1) In ZF (Zermelo–Fraenkel set theory without the axiom of choice), equivalent are: (i) every base of a second countable space has a countable subfamily which is a base; (ii) the axiom of countable choice fo...

In this note linearly ordered topological spaces (abbreviated LOTS) with a point-countable base are examined. It is shown that a LOTS is quasi-developable if and only if it has a <r-point-finite base and a LOTS with a point-countable base is paracompact. An example of a LOTS with a point-countable base that does not have a <r-point-finite base is given. Conditions are given for the metrizabilit...

Programming languages with countable nondeterministic choice are computationally interesting since countable nondeterminism arises when modeling fairness for concurrent systems. Because countable choice introduces non-continuous behaviour, it is well-known that developing semantic models for programming languages with countable nondeterminism is challenging. We present a step-indexed logical re...

We prove a strong convergence result for a sequence generated by Halpern's type iteration for approximating a common fixed point of a countable family of quasi-Lipschitzian mappings in a real Hilbert space. Consequently, we apply our results to the problem of finding a common fixed point of asymptotically nonexpansive mappings, an equilibrium problem, and a variational inequality problem for co...