Topology + Algebra and Analysis
32nd Summer Conference on Topology and Its Applications
For a limit ordinal λ, let (Aα)α < λ be a system of topological algebras (e.g., groups or vector spaces) with bonding maps that are embeddings of topological algebras, and put A = ∪α < λ Aα. Let (A, T) and (A, A) denote the direct limit (colimit) of the system in the category of topological spaces and topological algebras, respectively. One always has T ⊇ A, but the inclusion may be strict; however, if the tightness of A is smaller than the cofinality of λ, then A=T.
In 1988, Tkachenko proved that the free topological group F(X) is sequential when Xn is sequential, countably compact, and normal for every n. In particular, F(ω1) is sequential.
In this talk, we show that under the same conditions, the free topological vector space V(X) is sequential, and thus countably tight. Consequently, F(ω1) = colimα < ω1 F(α) and V(ω1) = colimα < ω1 V(α) not only as topological algebras, but also as topological spaces.
Copyright © 2017, the Authors
Lukács, Gábor and Dahmen, Rafael, "On the Tightness and Long Directed Limits of Free Topological Algebras" (2017). Summer Conference on Topology and Its Applications. 30.