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Let $X$ be a compact $T_1$ topological space consisting of more than one point, and suppose that $X$ is locally compact (i.e. every point has a local base of compact neighbourhoods), second countable, and is a Baire space. Does $X$ admit a non-constant continuous function into the real numbers?

[In C$^*$-algebra terms, let $A$ be a non-primitive separable C$^*$-algebra whose primitive ideal space is compact and $T_1$. Does the multiplier algebra of $A$ have non-trivial centre?]

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  • $\begingroup$ Why is the cofinite topology on $\Bbb N$ not a counterexample? $\endgroup$ – Henno Brandsma May 8 '20 at 15:38
  • $\begingroup$ @Henno Brandsma. I don't think that it is a Baire space. The intersection of all the co-singleton sets is empty. $\endgroup$ – Douglas Somerset May 8 '20 at 19:42
  • $\begingroup$ True, I see. Thx. $\endgroup$ – Henno Brandsma May 8 '20 at 19:43

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