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    $\begingroup$ Great answer. I would only note that there is a large area of computer science known as Domain Theory that tries to make the notion of decidability a la Turing the same as the notion of decidability given by considering maps into the Sierpinski space. Here is a short primer on these ideas: homepages.inf.ed.ac.uk/als/Teaching/MSfS/l3.ps $\endgroup$ Commented Dec 20, 2010 at 6:18
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    $\begingroup$ I believe the correct embedding theorem into powers of the Sierpinski space is the following: a topological space $X$ embeds into a product of copies of the Sierpinski space iff it is Kolmogorov (a.k.a. $T_0$). So the Sierpinski space does not quite have the universal role that you suggest. $\endgroup$ Commented Dec 24, 2010 at 9:30
  • $\begingroup$ @PeteL.Clark it depends on whether you really care about non-$T_0$ spaces. For all I care, every space is Tychonoff (a.k.a. completely regular or $T_{3\frac{1}{2}}$). $\endgroup$ Commented Jun 11, 2014 at 16:15
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    $\begingroup$ @David: I find your remark curious. If you only care about Tychonoff spaces, it follows that you don't care about the Sierpinski space. So why comment on it? $\endgroup$ Commented Jun 22, 2014 at 22:31
  • $\begingroup$ In c. 1996 Ī discovered this formulation of general topology for myself and was astonished that it’s not found in textbooks. BTW, this space is also attributed to P.S. Alexandrov. $\endgroup$ Commented Mar 22, 2015 at 15:52