Continuous map from function space to quotient space maps through projection?
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Suppose $F$ is a function space $Y^X$ with $Y$ discrete (so it has the topology of pointwise convergence), and $F'$ is another function space $Y'^{X'}$ with $Y'$ discrete, and suppose we have an action of some group $G$ on $X'$. We can extend this action to an action on $F'$ via $g.f' := (x' mapsto f'(g.x'))$. Let $pi : F' to F'/G$ be the quotient map of this action.
Is it true that any continuous map $theta : F to F'/G$ factors through $pi$? In other words, is it true that there is some continuous map $tildetheta : F to F'$ for which $theta = pi circ tildetheta$? Equivalently, is there a continuous choice of representatives $r_f in theta(f)$ for $f in F$?
I am not even sure about the case when $G$ acts regularly on $X'$ (i.e. $X'$ is a $G$-torsor), though I expect the answer to be "yes" in this case.
general-topology group-actions quotient-spaces
asked Nov 24 at 0:21
feralin
925717
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up vote
1
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Suppose $F$ is a function space $Y^X$ with $Y$ discrete (so it has the topology of pointwise convergence), and $F'$ is another function space $Y'^{X'}$ with $Y'$ discrete, and suppose we have an action of some group $G$ on $X'$. We can extend this action to an action on $F'$ via $g.f' := (x' mapsto f'(g.x'))$. Let $pi : F' to F'/G$ be the quotient map of this action.
Is it true that any continuous map $theta : F to F'/G$ factors through $pi$? In other words, is it true that there is some continuous map $tildetheta : F to F'$ for which $theta = pi circ tildetheta$? Equivalently, is there a continuous choice of representatives $r_f in theta(f)$ for $f in F$?
I am not even sure about the case when $G$ acts regularly on $X'$ (i.e. $X'$ is a $G$-torsor), though I expect the answer to be "yes" in this case.
general-topology group-actions quotient-spaces
asked Nov 24 at 0:21
feralin
925717
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Suppose $F$ is a function space $Y^X$ with $Y$ discrete (so it has the topology of pointwise convergence), and $F'$ is another function space $Y'^{X'}$ with $Y'$ discrete, and suppose we have an action of some group $G$ on $X'$. We can extend this action to an action on $F'$ via $g.f' := (x' mapsto f'(g.x'))$. Let $pi : F' to F'/G$ be the quotient map of this action.
Is it true that any continuous map $theta : F to F'/G$ factors through $pi$? In other words, is it true that there is some continuous map $tildetheta : F to F'$ for which $theta = pi circ tildetheta$? Equivalently, is there a continuous choice of representatives $r_f in theta(f)$ for $f in F$?
I am not even sure about the case when $G$ acts regularly on $X'$ (i.e. $X'$ is a $G$-torsor), though I expect the answer to be "yes" in this case.
general-topology group-actions quotient-spaces
asked Nov 24 at 0:21
feralin
925717
Suppose $F$ is a function space $Y^X$ with $Y$ discrete (so it has the topology of pointwise convergence), and $F'$ is another function space $Y'^{X'}$ with $Y'$ discrete, and suppose we have an action of some group $G$ on $X'$. We can extend this action to an action on $F'$ via $g.f' := (x' mapsto f'(g.x'))$. Let $pi : F' to F'/G$ be the quotient map of this action.
Is it true that any continuous map $theta : F to F'/G$ factors through $pi$? In other words, is it true that there is some continuous map $tildetheta : F to F'$ for which $theta = pi circ tildetheta$? Equivalently, is there a continuous choice of representatives $r_f in theta(f)$ for $f in F$?
I am not even sure about the case when $G$ acts regularly on $X'$ (i.e. $X'$ is a $G$-torsor), though I expect the answer to be "yes" in this case.
general-topology group-actions quotient-spaces
general-topology group-actions quotient-spaces
asked Nov 24 at 0:21
feralin
925717
asked Nov 24 at 0:21
feralin
925717
asked Nov 24 at 0:21
feralin
925717
asked Nov 24 at 0:21
feralin
925717
asked Nov 24 at 0:21
feralin
925717
925717
add a comment |
add a comment |
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