Let $X$ be Hausdorff and $\phi: G \times X \rightarrow X$ a continuous left action of $G$ on $X$. I would like to prove that the fixed point set $$X_H = \{ x \in X : h \cdot x = x \text{ for }h \in H\}$$ is closed for any subset $H$ of $G$.
Since $X_H$ is a subset of $X$, it is also Hausdorff and so the diagonal $$\Delta = X_H \times X_H$$ is closed. I know I have to somehow use the continuity of $\phi$, but I am now sure how to fit this in with $\Delta$ being closed. How would I proceed?
Note that $X_H = \bigcap_{h\in H} X_h$ and $X_h$ is closed since it's the set of those $x$ such that the maps $\text{Id}_X$ and $x\mapsto h\cdot x$ agree on.
That is, if $f, g:X\to Y$ are two maps such that $Y$ is Hausdorff then $\{x : f(x) = g(x)\}$ is closed since it's the preimage of the diagonal $\Delta_Y$ of $Y$ by the map $h = (f, g):X\to Y\times Y$, $h(x) = (f(x), g(x))$. Here we apply this proposition to $Y = X$ and $f = \text{Id}_X, g:x\mapsto h\cdot x$.