Is the quotient map on a locally compact topological group closed?

Question

Let $G$ be a locally compact group, $H$ a closed subgroup, $G/H$ the space of left cosets with the quotient topology, and $q:G\rightarrow G/H$ the projection map. Also, let $C(G,\mathbb{C})$ be the space of continuous functions from $G$ to $\mathbb{C}$. The author of a book I am reading is attempting to define a subspace of $C(G,\mathbb{C})$ using (among others) the condition "$q($supp$(f))$ is compact". Here is my problem: unless $q$ is a closed map I do not see how this condition can respect linearity (i.e., if $q($supp$(f))$ and $q($supp$(g))$ are compact then so is $q($supp$(f+g))$ for $f,g$ in this subspace). If $G$ is simply a topological group and $H$ is compact, then it is true that $q$ is closed. But in particular for locally compact groups can we say that $q$ is closed? My belief is that this is not true, but I could be wrong. Since the other conditions imposed on this "subspace" don't seem to help the matter, this would appear to be an error in the text.

Answer 1

Indeed, $q$ may not be closed. For instance, if $G=\mathbb{R}\times\mathbb{Z}$ and $H=\mathbb{Z}$ then you could have a closed subset of $G$ that contains points on infinitely many of the cosets of $\mathbb{R}$ which when projected down to $G/H\cong\mathbb{R}$ accumulate somewhere and thus the image is not closed. Presumably the intended condition is for $q(\operatorname{supp}(f))$ to be precompact (i.e. its closure is compact), rather than compact. (Or, perhaps the additional assumptions being made which you have not mentioned are used in order to conclude that $q(\operatorname{supp}(f+g))$ will be compact.)