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Let me start by reminding two constructions of topological spaces with such exotic combination of properties:

1) The elements are non-zero integers; base of topology are (infinite) arithmetic progressions with coprime first term and difference.

2) Take $\mathbb{R}^{\infty}\setminus \{0\}$ with product-topology and factorize by the relation $x\sim y \Leftrightarrow x=ty$ for some $t>0$ (infinite-dimensional sphere). Then consider only points with rational coordinates, all but finitely of them vanishing.

The first question is wether are these two examples homeomorphic or somehow related.

The second is an historical one. I've heard that the first example of such space belongs to P. S. Urysohn. What was his example?

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Just to be sure I get the example (2): would it be the same starting from $\mathbb{Q}^\infty$ with the product topology; then taking the subspace of sequences with all but finitely many vanishing coord.; then the set of all rays in it with the quotient topology? –  Pietro Majer Nov 23 '10 at 12:17
    
@Pietro: yes, it is the same (except isolated point 0 maybe) –  Fedor Petrov Nov 23 '10 at 12:56
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1 Answer

Urysohn's example of a countable connected Hausdorff space with a countable base was published in his last paper «Über die Mächtigkeit der zusammenhängenden Mengen», Math Annalen 94 (1925), 262—295.

Urysohn's original description of the space occupies about 4 pages so I would rather refrain from reproducing it here. This is probably the most complicated construction of its kind. Simpler examples were later obtained by Bing, Hewitt, Stone and others («Countable connected spaces» by Miller contains many relevant references).

There is also a Russian translation of Urysohn's paper in his collected works

  • Урысон П.С. Труды по топологии и другим областям математики [Том 1], ГИТТЛ, 1951, стр. 177-214.
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In example (2) the space is a quotient of a products of real lines, then the phrase "only points with rational coordinates" isnt well definited. May be you considering the point of the product with all but a finite vanish, and have distance $1$ from ${0}$? (i.e . the point on the 1-sphere by rational coordinates) Anyway here there is a (very nice) example of Bing: ams.org/journals/proc/1953-004-03/S0002-9939-1953-0060806-9/… –  Buschi Sergio Nov 22 '10 at 20:37
    
@Sergio: well, we may consider say "points with rational ratios of coordinates", it is maybe more rigorous, but actually does not matter. Thanks for the link! –  Fedor Petrov Nov 22 '10 at 20:58
    
@Andrey: thanks for the reference! –  Fedor Petrov Nov 22 '10 at 20:58
    
by the way, how may translation of 34-pages paper consist of 5 pages? –  Fedor Petrov Nov 22 '10 at 21:04
    
@Fedor: I have edited the answer. The example itself is described in Section 3 of the paper, pp. 189-193 in the Russian translation. –  Andrey Rekalo Nov 22 '10 at 21:24
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