$GL(n,\mathbb{C})$ is semi-locally simply connected.

Question

How can we show that $GL(n,\mathbb{C})$ is semi-locally simply connected.

Answer 1

$GL(n,\mathbb C)$ is an open set in the set of all $n\times n$ matrices with coefficients in $\mathbb C$. This set is homeomorphic to $\mathbb R^{2n^2}$. Now let $x\in GL(n,\mathbb C)$ so there exists an open ball $x\in B(x;\epsilon)\subset GL(n,\mathbb C)$. But the open ball $B(x;\epsilon)$ is contractible hence simply connected.