Two preservation results for countable products of sequential spaces

In the theory of Type Two Effectivity (Weihrauch 2000), computation on non-discrete spaces is performed by type two Turing machines, which compute with infinite words acting as names for elements of spaces. The connection between names and their associated elements is specified by a many-to-one relation called a representation. Such representations induce a topology on the named elements, namely the quotient topology of the relative product topology on the set of names. For certain admissible representations, the naming relations are themselves determined (up to a continuous equivalence) by the topology of the represented space. Moreover, the property of admissibility serves as a well-behavedness criterion for representations. In the case of the real numbers, for example, the property of admissibility exactly captures the distinction between reasonable computable representations (e.g. signed digit, Cauchy sequences with specified modulus, etc.), which are admissible, and unreasonable ones (e.g. ordinary binary/decimal notation), which are not. In general, one can argue that the spaces with admissible quotient representation are exactly the topological spaces that support a good (type two) computability theory, see (Weihrauch 2000; Schröder 2003). In his PhD thesis (Schröder 2003), the first author characterised those topological spaces which have admissible quotient representations as being exactly the T0 quotient spaces of countably-based spaces (qcb spaces). Qcb spaces are closed under many