# Continuous functions on a compact $T_1$ space

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?]

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