Topology

Abstract

Our matrix groups are all subsets of euclidean spaces, because they are all subsets of $$ {{\text{M}}_{\text{n}}}{\text{(R)}}\;{\text{ = }}\;{{\text{R}}^{{{\text{n}}^{\text{2}}}}}\;{\text{or}}\;{{\text{M}}_{\text{n}}}{\text{(C)}}\;{\text{ = }}\;{{\text{R}}^{{\text{2}}{{\text{n}}^{\text{2}}}}}\;{\text{or}}\;{{\text{M}}_{\text{n}}}{\text{(H)}}\;{\text{ = }}\;{{\text{R}}^{{\text{4}}{{\text{n}}^{\text{2}}}}}\;{\text{.}} $$ There are certain topological properties, notably connectedness and compactness, which some of our groups have and others do not. These properties are preserved by continuous maps and so are surely invariants under isomorphisms of groups. So a connected matrix group could not be isomorphic with a nonconnected matrix group, and a similar statement holds for compactness. We will define these properties and decide which of our groups have them. This will be done in sections B and C.KeywordsRational NumberOpen BallOpen SemeMaximal TorusQuadratic PolynomialThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.