Compact and compact Hausdorff

Summary

INTRODUCTION In this paper, separation axioms are not assumed without explicit mention; thus ‘compact’ means ‘quasi-compact’ in the sense of Bourbaki (every open cover has a finite subcover). Non-Hausdorff compact spaces arise, for example, in algebraic geometry (via the Zariski topology). In the summer of 1980, S. Eilenberg raised (orally) a question equivalent to the following: Is every compact space a quotient of some compact Hausdorff space? An affirmative answer, he said, would have interesting consequences. Unfortunately the answer is negative, as C.H. Dowker soon showed with the elegant example (hitherto unpublished) that follows. The present paper discusses various relations (some known, some new) between ‘compact’ and ‘compact Hausdorff’, suggested by Eilenberg's question and Dowker's answer to it.

EXAMPLE (Dowker) There exists a countable compact T₁ space D. satisfying the second axiom of countability, that is not a quotient of any compact Hausdorff space.

The space D consists of points a, b, c_ij (where i,j ε N = {1,2,3, …}), all distinct, topologized so that each c_ij is isolated, a neighbourhood base at a consists of the sets U_n = {c_ij: j≥ n} ⊂ {a}, nεN, and a neighbourhood base at b consists of the sets V_n = { c_ij: i ≥ n} ⊂ {b}, n ε N. Clearly this produces a compact T₁ topology; and D, being countable and first-countable, is also second-countable.