We prove that each non-separable completely metrizable convex subset of a Fréchet space is homeomorphic to a Hilbert space. This resolves a more than 30 years old problem of infinite-dimensional topology. Combined with the topological classification of separable convex sets due to Klee, Dobrowolski and Toruńczyk, this result implies that each closed convex subset of a Fréchet space is homeomorphic to $[0,1]ⁿ\times {[0,1)}^m\times \ell ₂\left(\kappa \right)$ for some cardinals 0 ≤ n ≤ ω, 0 ≤ m ≤ 1 and κ ≥ 0.

This paper begins the classification of topological actions on manifolds by compact, connected, Lie groups beyond the circle group. It treats multiaxial topological actions of unitary and symplectic groups without the dimension restrictions used in earlier works by M. Davis and W. C. Hsiang on differentiable actions. The general results are applied to give detailed calculations for topological actions homotopically modeled on standard multiaxial representation spheres. In the present topological...

We show that the strong dual X’ to an infinite-dimensional nuclear (LF)-space is homeomorphic to one of the spaces: ${ℝ}^{\omega }$, ${ℝ}^{\infty }$, $Q\times {ℝ}^{\infty }$, ${ℝ}^{\omega }\times {ℝ}^{\infty }$, or ${\left({ℝ}^{\infty }\right)}^{\omega }$, where ${ℝ}^{\infty }=lim{ℝ}^n$ and $Q={[-1,1]}^{\omega }$. In particular, the Schwartz space D’ of distributions is homeomorphic to ${\left({ℝ}^{\infty }\right)}^{\omega }$. As a by-product of the proof we deduce that each infinite-dimensional locally convex space which is a direct limit of metrizable compacta is homeomorphic either to ${ℝ}^{\infty }$ or to $Q\times {ℝ}^{\infty }$. In particular, the strong dual to any metrizable infinite-dimensional Montel space is homeomorphic either...

Let X be a topological group or a convex set in a linear metric space. We prove that X is homeomorphic to (a manifold modeled on) an infinite-dimensional Hilbert space if and only if X is a completely metrizable absolute (neighborhood) retract with ω-LFAP, the countable locally finite approximation property. The latter means that for any open cover $$𝒰$$ of X there is a sequence of maps (f n: X → X)nεgw such that each f n is $$𝒰$$ -near to the identity map of X and the family f n(X)n∈ω is locally finite...

We study the problem of approximating, up to homotopy, compact topological manifolds by real algebraic varieties. As a consequence, we realize any integral non-degenerate quadratic form as the intersection form of a real algebraic variety. This is related to a well-known result, due to Freedman [F], on the topology of closed simply-connected topological $4$-manifolds.

Let $L\left(X\right)$ be the space of all lower semi-continuous extended real-valued functions on a Hausdorff space $X$, where, by identifying each $f$ with the epi-graph $epi\left(f\right)$, $L\left(X\right)$ is regarded the subspace of the space ${Cld}_F^{*}(X\times ℝ)$ of all closed sets in $X\times ℝ$ with the Fell topology. Let $$LSC\left(X\right)=\{f\in L\left(X\right)\mid f\left(X\right)\cap ℝ\ne \varnothing ,f\left(X\right)\subset (-\infty ,\infty ]\}\text{and}{LSC}_B\left(X\right)=\{f\in L\left(X\right)\mid f\left(X\right)\text{is}\text{a}\text{bounded}\text{subset}\text{of}ℝ\}.$$ We show that $L\left(X\right)$ is homeomorphic to the Hilbert cube $Q={[-1,1]}^{\mathbb{N}}$ if and only if $X$ is second countable, locally compact and infinite. In this case, it is proved that $(L\left(X\right),LSC\left(X\right),{LSC}_B\left(X\right))$ is homeomorphic to $(ConeQ,Q\times (0,1),\Sigma \times (0,1\left)\right)$ (resp. $(Q,s,\Sigma )$) if $X$ is compact (resp. $X$ is non-compact), where $ConeQ=(Q\times 𝐈)/(Q\times \{1\left\}\right)$ is the cone over...

We study and classify topologically invariant σ-ideals with an analytic base on Euclidean spaces, and evaluate the cardinal characteristics of such ideals.

The focus of this paper are questions related to how various geometric and analytical properties of hyperbolic 3-manifolds determine the commensurability class of such manifolds. The paper is for the large part a survey of recent work.