1.1 Definitions
Let \(G\) be a topological group. We say \(G\) is a td group (also called l-group), if \(G\) is totally disconnected, Hausdorff (T2), and locally compact.
Let \(G\) be a topological group. We say that \(G\) has a compact open subgroup basis if \(\mathcal N(1)\) has a basis consisting of compact open subgroups.
Now we want to prove Theorem 13, i.e. the two definitions of td group are equivalent.
Let \(G\) be a topological group. If \(G\) has a compact open subgroup basis, then \(G\) is a nonarchimedean group.
Let \(G\) be a topological group. Then a neighborhood basis at unit \(1\) transports to a neighborhood basis at any \(x\in G\) by left translation.
A locally compact, Hausdorff, nonarchimedean group is totally disconnected.
Let \(G\) be a td group. Then every neighborhood of \(1\) contains a compact clopen neighborhood of \(1\).
Let \(G\) be a topological group. If \(K\) is compact and open subset of \(G\), then there exists \(V \in \mathcal N(1)\) such that \(K \cdot V \subseteq K\).
Let \(G\) be a td group. Inside a compact clopen neighborhood of \(1\), there exists a compact open subgroup.
Let \(G\) be a td group. Every neighborhood of \(1\) contains a compact open subgroup of \(G\).
If \(G\) is td group (i.e.totally disconnected, locally compact, and Hausdorff topological group), then \(G\) has a compact open subgroup basis.
If a topological group \(G\) has a compact open subgroup basis, then \(G\) is locally compact.
If a topological group \(G\) has a compact open subgroup basis, then \(G\) is totally disconnected.
Assuming \(G\) is Hausdorff, having a compact open subgroup basis is equivalent to being locally compact and totally disconnected.
Assuming \(G\) is Hausdorff, having a compact open subgroup basis is equivalent to being locally compact and nonarchimedean.