Prove it is locally connected topological group

Question

I got stuck on the following problem:

I am looking at the general linear group $GL_n(\Bbb R)$ which is a subset of $\Bbb R^{n^2}$. I want now to prove that $GL_n(\Bbb R)$ is a locally connected topological group, without the fact that it is locally compact.

Any help would be much appreciated! Thanks in advance.

Answer 1

This can be done in two steps:

1. $GL_n(\mathbb R)$, regarded as a subset of $\mathbb R^{n^2}$, is an open subset, as one proves using continuity of the determinant function $\text{det} : \mathbb R^{n^2} \to \mathbb R$
2. Every open subset of $\mathbb R^{n^2}$ is locally connected, as one proves using the standard "open ball" basis for the topology on $\mathbb R^{n^2}$.