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Let $(E, \mathcal{T})$ be a compact Hausdorff space. It is well known that every topology $\mathcal{U}$ coarser than $\mathcal{T}$ such that $(E, \mathcal{U})$ is Hausdorff is equal to $\mathcal{T}$.

Is the converse true?

(that is: if $\mathcal{T}$ is a coarsest topology amongst Hausdorff topology on $E$, then $(E, \mathcal{T})$ is compact)

Thanks in advance.

  • 1
    Your questio$n$ is probably sufficie$n$tly answered by the first comment. But a key word you might look for is "minimal Hausdorff space." $A$lso, note that some spaces don't _have_ "minimal Hausdorff" coarsenings -- the rational numbers for example.2013-11-15

1 Answers 1