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If $G$ is a topological group, $X$ is a topological space and $\theta:G\times X\to X$ is an action, then $X$ is called a $G$-space provided $\theta$ is continuous.

This is something I just read in a book: Suppose $G$ is a group, $X$ a topological space and $\theta :G\times X\to X$. Then if we give $G$ the discrete topology, then $X$ is a $G$-space because every $\theta$ is continuous when $G$ is discrete.

But I can't see why yet. If $G$ is discrete, I think $G\times X$ need not be discrete, so I don't see why every $\theta$ would be continuous. Or maybe if $O\subseteq X$ is open, could it be that $\theta^{-1}(O)$ is the product of $O$ with a subset of $G$? How can we see that?

Thank you.

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$G \times X$ is covered by the open sets $O_g = \{g\} \times X$ ($g \in G)$, and $\theta$ is continuous iff $\theta |_{O_g}$ is continuous for all $g$. Then we can say that the restriction of $\theta$ to $O_g$ is a bijection on $X$ (from the axioms of a group action). But we need this to be a continuous bijection. And this need not be true:

let $X$ be the Sorgenfrey line ($\mathbb{R}$ in the topology generated by all sets of the form $[a,b), a < b$)). Let $G = \{1,-1\}$ in the discrete topology (as a multiplicative group), and define $\theta(g,x) = gx$. Then for $g=-1$ we have that $x \rightarrow -x$ is not continuous on $X$, so $\theta$ is not a continuous action, although it is a valid group action.

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  • $\begingroup$ Thank you, I liked your answer very much! $\endgroup$ – JonSK Feb 20 '16 at 18:57
  • $\begingroup$ @JonSK Glad I could help! $\endgroup$ – Henno Brandsma Feb 20 '16 at 18:58

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