Show that the following are topological groups:
\((\mathbb{Z}, +)\)
\((\mathbb{R}, +)\)
\((\mathbb{R}_+, \cdot)\)
\((S^1, \cdot)\), where we take \(S^1\) to be the space of all complex numbers \(z\) for which \(\abs{z} = 1\).
The general linear group \(\mathrm{GL}(n)\), under the operation of matrix multiplication. (\(\mathrm{GL}(n)\) is the set of all nonsingular \(n\) by \(n\) matrices, topologized by considering it as a subset of euclidean space of dimension \(n^2\) in the obvious way.)
Solution
Each of the five spaces is a subspace of some \(\RR^k\), hence \(T_1\), and each is a group under the stated operation; it remains to show that multiplication and inversion are continuous.
1. (a) \((\ZZ,+)\) is a topological group.
The subspace topology on \(\ZZ\subseteq\RR\) is discrete, so \(\ZZ\times\ZZ\) is discrete and every map out of \(\ZZ\) or \(\ZZ\times\ZZ\) is continuous.
2. (b), (c) \((\RR,+)\) and \((\RR_+,\cdot)\) are topological groups.
Addition, negation, multiplication, and \(x\mapsto1/x\) on \((0,\infty)\) are continuous by E-YTG4V.
3. (d) \((S^1,\cdot)\) is a topological group.
Complex multiplication \((z,w)\mapsto zw\) and conjugation \(z\mapsto\bar z\) are continuous on \(\CC=\RR^2\), being polynomial in the real coordinates, and on \(S^1\) the inverse is \(z^{-1}=\bar z\); restrictions to \(S^1\) are continuous.
4. (e) \(\mathrm{GL}(n)\) is a topological group.
Each entry of \(AB\) is a polynomial in the entries of \(A\) and \(B\), so multiplication is continuous. By Cramer’s rule each entry of \(A^{-1}\) is a polynomial in the entries of \(A\) divided by \(\det A\ne0\), so inversion is continuous on \(\mathrm{GL}(n)\).
5. Q.E.D.
Steps 1 through 4 treat (a) through (e).