Is the set of irrationals a complete metric space?

[Eynstone]

I came across Alexandrov's theorem which says that if X is a Polish space then so is any Gδ subset of X. The set of irrationals appears to be a ground for suspicion : irrationals form a G-delta set of the reals & yet are not a complete metric space ( all under the usual metric).
There is, of course, a metric under which the irrationals are complete.Could someone clarify this? Thanks.

 

[Jamma]

From wiki:

"a Polish space is a separable completely metrizable topological space"

A completely metrizable space isn't the same as complete metric space. It means that there exists some metric on the space which induces the topology which is complete, so it seems that what you have said is totally correct.

 

[Bacle]

It seems like the thing to show is that the metric ρ(x,y)=d(x,y)+Ʃ_n=1^∞2^-nψ_n , with: ψ_n(x,y)=|f_i(x)-f_i(y)(y)|/[1+|f_i(x)-f_i(y)|]

And f_i(x) := 1/d(x,M-U_i) , where the irrationals are an intersection of the U_i

Generates the same topology as the standard one in the irrationals.

A tour-de-force ( or, like some say it, a tour-de-france ) of point-set topology.

 

[Bacle]

Just in case, the metric on the irrationals is described in Hocking and Young's Topology, in page 83-or-so. The book is a trip back in time where point-set topology was a big game. I find a lot of it interesting, but nowadays much of it seems like mor eof a curiosity; used in some areas (e.g., functional analysis, and assumed --and often swept under the rug-- in algebraic topology).

 

[Jamma]

Ah, ok. Was really scratching my head with that metric you described. Is it part of a more general technique?

 

[Bacle]

Yes, the proof that there exists a complete metric for a G_δ subset is constructive, and the metric given is the one I posted. It can be generalized for
other G_δ subsets , of course.

 

[Bacle]

Hope this is not too far off-topic , neither for this post nor the forum, but there are other results in the book one does not hear much about, like that of producing an actual metric to show that an inverse limit of metric spaces is a metric space. I guess that's the difference between clasical and modern topology; in classical, one can see better what's under the hood, in terms of underlying details, tho maybe the problem with classica is that of not being able to see the forest, from somuch detail.