Sequential topological conditions in in the absence of the axiom of choice

There are many topological results in Zermelo-Fraenkel set theory including the axiom of choice (ZFC) that are not true in the absence of choice, i. e. in ZF. Even if we restrict our attention to many “familiar” topological results are not provable in ZF, although in most cases their validity follows from a weaker version of the axiom of choice, CC( ). Definition 0.1 The axiom of countable choice (CC) states that every countable family of non-empty sets has a choice function. Definition 0.2 CC( ) is the axiom of countable choice restricted to families of sets of real numbers. Under CC( ), it is known to be true that “a subspace of is compact if and only if it is sequentially compact” (see [4, p. 128]). In this paper, we will investigate under which conditions this equivalence remains valid and we will exhibit a list of equivalent conditions to this one. In [4], Felscher has a partial answer to this question by saying that CC( ) is equivalent to this condition together with the idempotence of the sequential closure in . We will see that CC( ) is not necessary to prove the equivalence between compact and sequentially compact subspaces of , after proving it from a weaker form of choice. The equivalence is not provable in ZF, once that implies that “every Dedekind-finite subset of is finite”, known to be not provable in ZF (basic Cohen model). We call a set finite if it is either empty or equipollent to a natural number. Otherwise the set is infinite. Definition 0.3 A set is Dedekind-finite if no proper subset of is equipollent to . Otherwise is Dedekind-infinite. Proposition 0.4 A set is Dedekind-infinite if and only if it has a countable subset i. e., there is an injection from to . Throughout this paper we work in ZF, the Zermelo-Fraenkel set theory without axiom of choice.