Example 3

If G is a topological group, its topology is compatible with the structure of the group $G^0$ which is the opposite of G; $G^0$, with this topology, is said to be the topological group opposite to G.

Axioms $(\mathrm{GT}_\mathrm{I})$ and $(\mathrm{GT}_\mathrm{II})$ are equivalent to the following:

(GT'). The mapping $(x, y) \to xy^{-1}$ of $G \times G$ into $G$ is continuous.

Clearly $(\mathrm{GT}_\mathrm{I})$ and $(\mathrm{GT}_\mathrm{II})$ together imply (GT'). Conversely, (GT') implies $(\mathrm{GT}_\mathrm{II})$, for $x \to ex^{-1} = x^{-1}$ is then continuous; and (GT') and $(\mathrm{GT}_\mathrm{II})$ together imply $(\mathrm{GT}_\mathrm{I})$, for $(x, y) \to x(y^{-1})^{-1} = xy$ is then continuous.

If $a$ is any element of $G$, the left translation $x \to ax$ (resp. the right translation $x \to xa$) is continuous, by $(\mathrm{GT}_\mathrm{I})$, and is therefore a homeomorphism of $G$ onto $G$. The mappings $x \to axb$, as $a$ and $b$ run through $G$, thus form a group of homeomorphisms of $G$; the mappings $x \to axa^{-1}$ (resp. $x \to ax, x \to xa$), where $a$ runs through $G$, form a subgroup of this group of homeomorphisms. Again, since the symmetry $x \to x^{-1}$ is an involutory permutation of $G$, axiom $(\mathrm{GT}_\mathrm{II})$ shows that this mapping is a homeomorphism of $G$ onto $G$.

If $A$ is an open (resp. closed) subset of $G$, and if $x$ is any point of $G$, then the sets $x.A, A.x$ and $A^{-1} (*)$ are open (resp. closed), for they are the transforms of $A$ under one of the preceding homeomorphisms. If $A$ is open and $B$ is any subset of $G$, then $AB$ and $BA$ are open, because they are unions of open sets [axiom $(\mathrm{O}_\mathrm{I})$]. If $V$ is a neighbourhood of $e$ in $G$, then $VA$ and $AV$ are neighbourhoods of $A$; for if $W$ is an open neighbourhood of $e$ contained in $V$, then $WA$ and $AW$ are open and contain $A$.

On the other hand, $AB$ need not be closed when $A$ is closed, even if $B$ is closed too (cf. § 4, no. 1, Corollary 1 to Proposition 1).

\* For example, in the additive group of the real line $\mathbf{R}$, the subgroup $\mathbf{Z}$ of the rational integers is closed, and so is the subgroup $\theta \mathbf{Z}$ consisting of all integer multiples $n\theta$ of an irrational number $\theta$; but the subgroup $\mathbf{Z} + \theta \mathbf{Z}$ of $\mathbf{R}$, which is the set of all real numbers $m + n\theta$ (where $m$ and $n$ take all integer values) is not closed in $\mathbf{R}$, as we shall see in Chapter V, § 1.

Again, let $A$ be the subset of the additive group of $\mathbf{R} \times \mathbf{R}$ which consists of all $(x, y)$ pairs such that $x \geqslant 0$ and $0 \leqslant y \leqslant 1 - \frac{1}{x+1}$; and let $B$ be the set of all pairs $(x, 0)$, as $x$ runs through $\mathbf{R}$. $A$ and $B$ are closed, but $A + B$ is the set of all pairs $(x, y)$ such that $0 \leqslant y < 1$, and is not closed in $\mathbf{R} \times \mathbf{R}$. \*

Let $X$ be a topological space and let $f$ and $g$ be two mappings of $X$ into a topological group $G$. If $f$ and $g$ are continuous at a point $x_0$

(*) We recall that if $A$ and $B$ are two subsets of a group $G$, then $A.B$ or $AB$ denotes the set of all products $xy$ where $x \in A$ and $y \in B$; $A^{-1}$ denotes the set of all elements $x^{-1}$ where $x \in A$. If $B$ consists of a single element $x$, we write $x.A$ or $xA$ (resp. $A.x$ or $Ax$) in place of $\{ x \}.A$ (resp. $A.\{ x \}$).

of X, then so are $f^{-1}$ and $fg$(*), by Theorem 2 of Chapter I, § 2, no. 1. In particular, the continuous mappings of X into G form a subgroup of the group $G^X$ of all mappings of X into G.

Again, let $f$ and $g$ be two mappings of a set X, filtered by a filter $\mathfrak{F}$, into a Hausdorff topological group G. If $\lim_{\mathfrak{F}} f$ and $\lim_{\mathfrak{F}} g$ exist, then so do $\lim_{\mathfrak{F}} f^{-1}$ and $\lim_{\mathfrak{F}} fg$, and we have (Chapter I, § 7, no. 4, Proposition 9, Corollary 1)

(1)

(2)

When G is a commutative group, written additively, the axiom (GT') indicates that $(x, y) \to x - y$ is a continuous mapping. If $f$ and $g$ are mappings of a topological space X into G, continuous at a point $x_0$, then $f - g$ is continuous at this point. The formulas (1) and (2) can be transcribed similarly.