Exercise 1288Topology › Topology of Mn(ℝ)L1Added 4 Nov 2026

Interior and boundary of $GL_n(\mathbb{R})$

#interior#boundary#GL_n

Statement

Determine the interior and the boundary of $GL_n(\mathbb{R})$ in $M_n(\mathbb{R})$ .

Hint

$GL_n$ is open; its boundary is the set of singular matrices.

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Solution

Interior. Since $GL_n$ is open, $\mathring{GL_n}=GL_n$ .

Closure. Every singular matrix $A$ is a limit of invertible matrices: $A+\varepsilon I$ is invertible for small $\varepsilon>0$ . So $\overline{GL_n}=M_n(\mathbb{R})$ .

Boundary. $\partial GL_n=\overline{GL_n}\setminus\mathring{GL_n}=M_n(\mathbb{R})\setminus GL_n$ , the set of singular matrices.

Conclusion. Interior $=GL_n$ ; boundary $=$ singular matrices.

Interior and boundary of $GL_n(\mathbb{R})$

Statement

Solution

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