#### Document Type

Topology + Algebra and Analysis

#### Publication Date

6-2017

#### Publication Source

32nd Summer Conference on Topology and Its Applications

#### Abstract

For a limit ordinal λ, let (A_{α})_{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} A_{α}. 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 X^{n} is sequential, countably compact, and normal for every n. In particular, F(ω_{1}) 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(ω_{1}) = colim_{α < ω1} F(α) and V(ω_{1}) = colim_{α < ω1} V(α) not only as topological algebras, but also as topological spaces.

#### Copyright

Copyright © 2017, the Authors

#### eCommons Citation

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.

https://ecommons.udayton.edu/topology_conf/30

## Comments

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