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A unified approach, that uses countable sets with convergences and basics of the theory of LF-spaces, is given to some old results of the Silverman–Toeplitz type obtained in the late 1930's by H.J. Hamilton for matrix transformations of multiple sequences.

Suppose M is a non-compact connected n -manifold without boundary, D ( M ) is the group of C ∞ -diffeomorphisms of M endowed with the Whitney C ∞ -topology and D 0 ( M ) is the identity connected component of D ( M ) , which is an open subgroup in the group D c ( M ) ⊂ D ( M ) of compactly supported diffeomorphisms of M . It is shown that D 0 ( M ) is homeomorphic to N × R ∞ for an l 2 -manifold N whose topological type is uniquely determined by the homotopy type of D 0 ( M ) . For instance, D 0 ( M ) is homeomorphic to l 2 × R ∞ if n = 1 , 2 or n = 3 and M is orientable and irreducible. We also show that for any compact connected n -manifold N with non-empty boundary ∂ N the group D 0 ( N ∖ ∂ N ) is homeomorphic to D 0 ( N ; ∂ N ) × R ∞ , where D 0 ( N ; ∂ N ) is the identity component of the group D ( N ; ∂ N ) of diffeomorphisms of N that do not move points of the boundary ∂ N .

Let X be a locally compact Polish space and G a non-discrete Polish ANR group. By C(X,G), we denote the topological group of all continuous maps f:X \to G endowed with the Whitney (graph) topology and by C_c(X,G) the subgroup consisting of all maps with compact support. It is known that if X is compact and non-discrete then the space C(X,G) is an l_2-manifold. In this article we show that if X is non-compact and not end-discrete then C_c(X,G) is an (R^\infty \times l_2)-manifold, and moreover the pair (C(X,G), C_c(X,G)) is locally homeomorphic to the pair of the box and the small box powers of l_2., 11 pages

In this paper, we prove that an L-topological vector space is locally convex (locally bounded) iff its induced I(L)-topological vector space is locally convex (locally bounded), where I(L) is the L-fuzzy unit interval. Also, we give a necessary and sufficient condition for the induced I(L)-topological vector space to be I(L)-fuzzy normable.

In this paper, the relation between two definitions of locally convex I -topological vector spaces is studied. These two definitions are introduced by Katsaras [Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984) 143–154] and Wu and Li [Convexity and fuzzy topological vector spaces, Science Exploration (China) 4(1) (1984) 1–4], respectively. We prove that the locally convex I -topological vector space in the sense of Wu and Li is a special case of that in the sense of Katsaras. We introduce the notion of generalized fuzzy semi-norm and prove that every locally convex I -vector topology in the sense of Katsaras on X can be determined by a family of generalized fuzzy semi-norms on X .

For all linear and many nonlinear operators between Frechet spaces, several continuity results are established. Especially, a new closed graph theorem is given.

In this paper, using the theory of L-topological vector spaces, we study some properties of inductive topologies determined by the family of L-fuzzy linear order-homomorphisms defined by Fang Jin-xuan (J. Fuzzy Math. 4(1) (1996) 93), and give a characterization of inductive topologies determined by a single L-fuzzy linear order-homomorphism. As an example, we point out the quotient L-topology of L-tvs is an inductive topology.

We shall construct in ZFC two Frechet–Urysohn α 4 -spaces, the product of which is α 4 , but fails to be Frechet–Urysohn. This answers Nogura's question from 1985.

We prove that for a metrizable space X the following are equivalent: (i) the free Abelian topological group A(X) is the inductive limit of the sequence {A n (X):n∈ N } , where A n (X) is formed by all words of reduced length ≤n ; (ii) X is locally compact and the set of all non-isolated points of X is separable. In the non-Abelian case, for a metrizable X the following are equivalent: (i) the free topological group F(X) is the inductive limit of the sequence {F n (X):n∈ N } ; (ii) X is either locally compact separable or discrete.