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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.

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    Your first set contains a fragment of a formula, and is not correct as written. Perhaps you meant to write $$F=\bigl\{(x,gx)\in X\times X\mid x\in X, g\in G\bigr\}\ ?$$ Or maybe $$F=\bigl\{(x,y)\in X\times X\mid \exists g\in G (y=gx)\bigr\}\ ?$$2011-12-04
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    @Arturo Magidin : they seem the same!! actually i mean the set $F=\{(x,y)\in X\times X\; |\;orb(x)=orb(y)\}$2011-12-04
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    The two I wrote are the same; the one you wrote is not correct use of mathematical quantifiers. "For all $x$" etc are prefixes, not suffixes and not sentences in and of themselves. We (should) never write "$(a+b)+c = a+(b+c)\ \forall a,b,c$", we should write $\forall a,b,c \bigl( (a+b)+c = a+(b+c)$". What you wrote originally was the equivalent of an opening clause "Where for all $x\in X$ and all $g\in G$ the following holds..." and then you never said what "the following" was.2011-12-04

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