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SHEILA CARTER AND F. J. CRAVEIRO DE CARVALHO
i2,
if x = i1 ifx∈(1,x1]
z −1
1 (x−1)+1,
f(x) =
xi+1 −xi −wi+1+wi
ifx∈[xi,xi+1],i∈N (x−yi+1)+wi+1, ifx∈[yi+1,yi],i∈N
x1−1
zi+1−zi (x−xi)+zi,
−yi+1 +yi w1−y1+x,
ifx∈[y1,+∞)
4. T0 Spaces
For T0 spaces the only T0 minimal universal space is the Sierpinski space.
Theorem 4.1. There exist a subset Y of R and a group G of homeomorphisms of RY such that RY /G is the Sierpinski space.
Proof. Fixp1,...,pn inR. TakeY =R\{p1,...,pn}andletGbethegroupof homeomorphisms of RY which fix p1, . . . , pn. The topology for RY /G is formed by all the subsets which contain pY (Y ). If n = 1 that space is the Sierpinski space.
5. Concluding Remarks
We assumed above Y to be non-empty. Obviously if Y is empty then R = RY .
For X as in 1), if we take for G the group of homeomorphisms f such that f(0) = 0 and irrationals (resp. non-zero rationals) are mapped to irrationals (resp. non-zero rationals) then X and R/G are homeomorphic. Again we refer to [2].
The case X as in 2) was dealt with in [1] as we have already mentioned.
Clearly, if X is as in 3) then it is not homeomorphic to a space R/G since it is not connected.
Finally, the Sierpinski space can be obtained as R/G taking, for instance, the group of homeomorphisms of R that fix a given point, as it was pointed out before.
References
[1] Sheila Carter and F. J. Craveiro de Carvalho, Homeomorphisms of R and the Davey space, Appl. Gen. Topol. (to appear).
[2] F. J. Craveiro de Carvalho and Bernd Wegner, A line thicker than the Michael line, “The J. A. Pereira da Silva Birthday Schrift”, Textos de Matem´atica 32, DMUC, Coimbra, 2002.
[3] K. D. Magill, Jr., Universal topological spaces, Amer. Math. Monthly 95 (1988), no. 10,
942-946.
[4] E. A. Michael, The product of a normal space and a metric space need not be normal, Bull.
Amer. Math. Soc. 69 (1963), 375-376.
