let $X$ be a top space and $G$ a group acting on $X$. Consider $ F=\bigl\{(x,gx)\in X\times X\mid x\in X, g\in G\bigr\}\ $ i want to write an homeomorphic image of $F$.
for example take $G=\mathbb Z_2=\{1,-1\}$ acting on the sphere $S^d$ by multiplication. in This case $F=\{(x,y)\in X \times X \;| \; x=\pm y\}$ so we can write it as the disjoint union $F=F_1\sqcup F_2$ where $F_1=\{(x,x)\in S^d\times S^d\}\cong diagonal(S^d\times S^d)\cong S^d$ and $F_2=\{(x,-x)\in S^d\times S^d\}\cong S^d$ Hence $F$ is homeomorphic to a disjoint union of two copies of $S^d$,i.e. $F=S^d\sqcup S^d$
Is there a similar way to write $F$ for more general $X$ and $G$? thanks for help.