$GL_{n+1}(\mathbb R)/GL_n(\mathbb R)$ is not compact

Question

Let $GL_{n+1}(\mathbb R)\subset M_{n+1}(\mathbb R )$ be given the subspace topology and identify $GL_n(\mathbb R)$ with the subset of $GL_{n+1}(\mathbb R)$ consisting of the matrices of the form $\begin{bmatrix}1&0 \\ 0&A \end{bmatrix}$ with $A\in GL_n(\mathbb R)$ .Then I have to show that $GL_{n+1}(\mathbb R)/GL_n(\mathbb R)$ is not compact in the quotient topology.

I understand that $GL_{n+1}(\mathbb R)$ is not compact itself,because it is not bounded.But I have no clear idea about what happens when I mod out by the subset mentioned above.

Any help would be appreciated.Thanks in advance.

Answer 1

The function $$B\in GL_{n+1}(\mathbb R)\mapsto b_{11}\in\mathbb R$$ is continuous and constant on equivalent classes, so it factors thru the quotient giving a continuous unbounded function. Therefore the quotient is not compact.