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Let $G$ be a topological group acts on the topological space $X$, for an elememt $g\in G$, let's define the map $f:X\rightarrow X, f(x)=g*x$.

I am trying to find if $f$ is continuous?

my best thanks

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Without further condition on the group action, $f_g$ is not necessarily continuous. Usually, with topological groups, you specify that the group action: $\phi:G\times X\rightarrow X$ is continuous. Then $f_g(x)=\phi(g,x)$ is continous. – Thomas Andrews Oct 19 '12 at 0:09
yes, actually $\phi:G\times X \rightarrow X$ is continuous, but how can we prove that $f_{g}:X\rightarrow X$ defined above is continuous – kiranovalobas Oct 19 '12 at 0:16
Because $h_g:X\to G\times X$ defined as $h_g(x)=(g,x)$ is continuous. And $f_g=\phi \circ h_g$ – Thomas Andrews Oct 19 '12 at 0:22

1 Answer 1

When we say $G$ acts on some object $X$, we usually mean that each group element $g$ is assigned a map $f_g:X\to X$. The nature of this map depends on the context of the object. We should ask: To which category does $X$ belong? The category determines what the objects and maps are, and how maps are composed with one another. If $X$ is a set, the maps are ordinary functions. If $X$, however, is a topological space, the maps we are interested in are usually the continuous maps. So when $G$ acts on a topological space $X$, it is usually assumed that each $f_g$ is continuous.

This is automatic if the group action is continuous, meaning that the map $G\times X\xrightarrow \mu X, (g,x)\mapsto f_g(x)$, is continuous. In that case, for a fixed $g\in G$, the function $f_g$ is the composite $\mu\circ\text{in}_g$, where $\text{in}_g(x)=(g,x)$.

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