As a manifold, GL(n, R) is not connected but rather has two connected components: the matrices with positive determinant and the ones with negative determinant. The identity component, denoted by GL⁺(n, R), consists of the real n×n matrices with positive determinant. ... The group GL(n, R) is also noncompact.

Is GL 2 R connected?

The Gram-Schmidt process allows us to make a deformation retract from GL+(2,R) to SO(2). Since SO(2)≃S1, it follows that SO(2) is path-connected and thus GL+(2,R) must also be.

Is the general linear group closed?

Every linear algebraic group is a closed subgroup of GL(n,R) (that is, closed as a subset of the space GL(n,R) with the Zariski topology).