I will have to think more about the issue of continuity vs uniform continuity. I suppose my last remaining argument would be the fact that Bishop—Bridges’ classic book on constructive analysis uses uniform continuity on bounded sets rather than continuity, which suggests that it is probably better for constructive analysis at least. But maybe they did not analyze the issue carefully enough, or maybe the relevant issues here are for some reason different.
To fix the argument that every locally compact Polish space admits a proper metric, let f be as before and let F(x,y)=∞ if d(x,y)≥f(x) and F(x,y)=f(x)/[f(x)−d(x,y)] if d(x,y)<f(x). Next, let g(y)=minn[n+F(xn,y)], where (xn) is a countable dense sequence. Then g is continuous and everywhere finite. Moreover, if S=g−1([0,N]), then S⊆⋃n≤NB(xn,(1−1/N)f(xn)) and thus S is compact. It follows that the metric d′(x,y)=d(x,y)+|g(y)−g(x)| is proper.
Hm, perhaps I should figure out what the significance of uniform continuity on bounded sets is in constructive analysis before dismissing it, even though I don’t see the appeal myself, since constructive analysis is not a field I know much about, but could potentially be relevant here.
f is the reciprocal of what it was before, but yes, this looks good. I am happy with this proof.
I will have to think more about the issue of continuity vs uniform continuity. I suppose my last remaining argument would be the fact that Bishop—Bridges’ classic book on constructive analysis uses uniform continuity on bounded sets rather than continuity, which suggests that it is probably better for constructive analysis at least. But maybe they did not analyze the issue carefully enough, or maybe the relevant issues here are for some reason different.
To fix the argument that every locally compact Polish space admits a proper metric, let f be as before and let F(x,y)=∞ if d(x,y)≥f(x) and F(x,y)=f(x)/[f(x)−d(x,y)] if d(x,y)<f(x). Next, let g(y)=minn[n+F(xn,y)], where (xn) is a countable dense sequence. Then g is continuous and everywhere finite. Moreover, if S=g−1([0,N]), then S⊆⋃n≤NB(xn,(1−1/N)f(xn)) and thus S is compact. It follows that the metric d′(x,y)=d(x,y)+|g(y)−g(x)| is proper.
Anyway I have fixed the typo in my previous post.
Hm, perhaps I should figure out what the significance of uniform continuity on bounded sets is in constructive analysis before dismissing it, even though I don’t see the appeal myself, since constructive analysis is not a field I know much about, but could potentially be relevant here.
f is the reciprocal of what it was before, but yes, this looks good. I am happy with this proof.