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Open subsets of LF-spaces

Mine K., Sakai K.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

Vol. 56, no 1

25-37

Let F = ind lim Fn be an infinite-dimensional LF-space with density dens F = r ( ≥ ℵo) such that some Fn is infinite-dimensional and dens Fn = r. It is proved that every open subset of F is homeomorphic to the product of an l2(r)-manifold and R∞ = ind lim Rn (hence the product of an open subset of l2(r) and R∞). As a consequence, any two open sets in F are homeomorphic if they have the same homotopy type.

Hyperspaces of finite sets in universal spaces for absolute Borel classes

Mine K., Sakai K., Yaguchi M.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

Vol. 53, no 4

409-419

By Fin(X) (resp. Fink (X)), we denote the hyperspace of all non-empty finite subsets of X (resp. consisting of at most k points) with the Vietoris topology. Let ℓ2 (τ) be the Hilbert space with weight τ and ℓf2 (τ) the linear span of the canonical orthonormal basis of ℓ2 (τ). It is shown that if E = ℓf2 (τ) or E is an absorbing set in ℓ2 (τ) for one of the absolute Borel classes aα (τ) and Mα (τ) of weight ≤ τ (α > 0) then Fin(E) and each Fink (E) are homeomorphic to E. More generally, if X is a connected E-manifold then Fin(X) is homeomorphic to E and each Fink (X) is a connected E-manifold.