[Paper Review] On topological groups containing a Fréchet-Urysohn fan

Тарас Банах|arXiv (Cornell University)|Nov 20, 2009

Advanced Topology and Set Theory19 citations

TL;DR

This paper investigates topological groups containing a Fréchet-Urysohn fan (space V), proving that if such a group is a perfectly normal sequential space or a normal k-space, then every closed metrizable subset must be locally compact. The key result establishes that non-metrizable topological groups which are direct limits of closed metrizable subsets are homeomorphic to the product of a $k_{\omega}$-space and a discrete space, thereby characterizing their topological structure and ruling out certain algebraic embeddings.

ABSTRACT

Suppose G is a topological group containing a (closed) topological copy of the Frechet-Urysohn fan. If G is a perfectly normal sequential space (a normal k-space) then every closed metrizable subset in $G$ is locally compact. Applying this result to topological groups whose underlying topological space can be written as a direct limit of a sequence of closed metrizable subsets, we get that every such a group either is metrizable or is homeomorphic to the product of a $k_ω$-space and a discrete space.

Motivation & Objective

• To determine topological conditions under which a space cannot support a topological group or convex structure.
• To analyze the implications of a topological group containing a closed copy of the Fréchet-Urysohn fan (space V).
• To characterize the structure of topological groups that are direct limits of closed metrizable subsets.
• To prove that such groups are either metrizable or homeomorphic to a product of a $k_{\omega}$-space and a discrete space.
• To establish that the presence of both test spaces $K$ and $V$ forbids a space from being a closed multiplicative subset of a topological group or a closed convex set in a linear topological space.

Proposed method

• Define two test spaces: $K$ (a non-locally compact metrizable space) and $V$ (the Fréchet-Urysohn fan), as obstructions to algebraic structures.
• Use the direct limit topology on $V$ to model spaces like $l_2^\infty$, which arise as direct limits of nested metrizable subspaces.
• Construct a map $f: K \times V \to X$ representing group multiplication or convex combination, and analyze its continuity and closed embedding properties.
• Prove that if $X$ contains both $K$ and $V$ as closed subspaces, then certain sets like $Z = \{f(x_{n,m}, y_{n,k(n,m)})\}$ are closed in $X$, leading to a contradiction if $X$ supports a group or convex structure.
• Use the $k$-space property and compactness arguments to show that $f(x_0, y_0)$ is not in the closure of such a set, contradicting continuity.
• Apply the structure theorem for $\mathcal{M}_\omega$-spaces to show that non-metrizable groups in this class are homeomorphic to $H \times D$, where $H$ is a $k_\omega$-space and $D$ is discrete.

Experimental results

Research questions

• RQ1 Under what topological conditions does a space fail to support a topological group structure?
• RQ2 Can a space containing a closed copy of the Fréchet-Urysohn fan and the space $K$ be homeomorphic to a closed multiplicative subset of a topological group?
• RQ3 What structural constraints does the presence of a Fréchet-Urysohn fan impose on a topological group that is a direct limit of closed metrizable subspaces?
• RQ4 Is the direct limit of a sequence of closed metrizable subsets in a topological group necessarily homeomorphic to a product of a $k_\omega$-space and a discrete space if not metrizable?
• RQ5 Does every closed metrizable subset in a normal $k$-space containing $V$ and $K$ have to be locally compact?

Key findings

• If a normal $k$-space contains closed copies of both $K$ and $V$, it cannot be homeomorphic to any closed multiplicative subset of a topological group.
• If a normal $k$-space contains a closed copy of $V$, then every closed metrizable subset in it must be locally compact.
• A topological group that is an $\mathcal{M}_\omega$-space and not metrizable must contain a closed copy of the Fréchet-Urysohn fan $V$, and every closed metrizable subset is locally compact.
• Every non-metrizable topological group that is an $\mathcal{M}_\omega$-space is homeomorphic to the product of a $k_\omega$-space and a discrete space.
• An open separable subgroup $H$ of such a group is a $k_\omega$-space, and the whole group decomposes as $H \times D$ for some discrete space $D$, implying the group is locally $k_\omega$.