Dimension, metrics and computability in metric spaces

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We considered the behavior of (transfinite) dimension functions on finite (and countable) unions of closed subsets in separable metrizable spaces. Let K be a class of spaces, α,β ordinal numbers with β<α and X be a space belonging to K with dX=α, where d denotes a dimension function monotonic for closed subspaces. Put m(X,d,β,α)=min{k:X is a union of k-many closed subspaces X1,…, Xk with d(Xk)≦β}, mK(d,β,α)=min(m(X,d,β,α):X is in K} and MK(d,β,α)=sup{m(X,d,β,α):X is in K}. Every ordinal number α can be represented as α=λ(α)+n(α),where λ(α) is a limit ordinal and n(α) is a natural number. For ordinal numbers β<α we put p(β,α)=(n(α)+1)/(n(β)+1) and q(β,α)=the smallest integer ≧p(β,α). Then we have the following : (1)Let P be the class of separable completely metrizable spaces. If a and B are finite ordinals with β<α, then mP(Cmp, β,α)=q(β,α) and MP(Cmp,β,α)=∞. (2)If α and β are infinite ordinals with β<α, then mP(trInd,β,α)=q(β,α) if λ (β) =λ(α), mP(trInd,β,α) does not exist if λ(β)≠λ(α) and MP(trInd,β,α)=∞ if λ(β)=λ(α), MP(trInd,β,α) does not exist if λ(β)≠(α). We also proved that there is no upper bound of trcmp in metrizable spaces. Concern with the theoretical computer science, we proved that the Lawson topology of the space of formal balls in the Hilbert space coincides with the product topology, but the Lawson topology of the space of formal balls in the space of real-valued functions on the closed interval does not coincide with the product topology. We also have several results about topological dyanamics, cohomological dimension of compact metric spaces, ordinal differential equations, and geometry.