Detecting topological groups which are (locally) homeomorphic to LF-spaces

Abstract

We prove that a topological group G is (locally) homeomorphic to an LF-space if G = n∈ω Gn for some increasing sequence of subgroups (Gn)n∈ω such that (1) for any neighborhoods Un ⊂ Gn, n ∈ ω, of the neutral element e ∈ Gn ⊂ G, the set(2) each group Gn is (locally) homeomorphic to a Hilbert space;(3) for every n ∈ N the quotient map Gn → Gn/G n−1 is a locally trivial bundle; (4) for infinitely many numbers n ∈ N each Z-point in the quotient space Gn/G n−1 = {xG n−1 : x ∈ Gn} is a strong Z-point.2010 Mathematics Subject Classification. 57N20; 46A13; 22A05.

On homeomorphism groups of non-compact surfaces, endowed with the Whitney topology

“…This implies that M contains a handle or a Klein bottle with a hole. But this contradicts Lemma 9 (1), (2). Now, assume that M n has two boundary circles C 1 , C 2 meetingM n .…”

“…Following [2], we say that a topological group G carries the strong topology with respect to a tower of subgroups…”

“…In this section we recall a criterion for recognizing topological groups homeomorphic to R ∞ × l 2 , which is proved in [2] and is based upon the topological characterization of the space R ∞ × l 2 given in [1]. First we recall some necessary definitions.…”

A topological characterization of LF-spaces

“…In this paper we shall present a simple criterion for recognizing topological spaces that are homeomorphic to (open subspaces of) LF -spaces. This criterion has been applied in [3], [4] and [7] for detecting topological groups that are homeomorphic to (open subspaces of) LF-spaces.…”

Completeness of Infinite-dimensional Lie Groups in Their Left Uniformity

“…In[3], (G, O) is then said to carry the strong topology.4 If e = x ∈ G n , there is an open identity neighbourhood V ⊆ G n with x ∈ V . Let W ⊆ G be an open identity neighbourhood such that W ∩ G n = V .…”