0

Let $f$ be a continuous mapping from $(\mathbb [0,1],\mathcal T_{st})$ to $(\mathbb N,\mathcal T_c)$, where $\mathcal T_{st}$ and $\mathcal T_c$ indicate, respectively, the standard topology and the cofinite topology. Prove $f$ is constant, that is, its image is a singleton.

I have thought about this a lot, but have gotten nowhere. I suspect Baire's theorem should come into play. Can anyone please help?

giobrach
  • 7,490

1 Answers1

1

For any natural $n$ the set $\{n\}$ is closed, and so $F_n=f^{-1}(n)$ is a sequence of closed mutually disjoint sets, which cover $[0,1]$. By Sierpiński theorem (see it for example in Engelking's General Topology), At most one of $F_n$ is nonempty, and so the image of $f$ contains just one point.

erz
  • 845
  • Nice! Do you think there's a way to prove this by using density properties in place of connectedness? – giobrach May 09 '17 at 00:40
  • I'm not sure what you mean, but take a compact set with countably many components, and just map each of them into different natural numbers. Preimage of any finite set will be the union of finitely many components, and so the map is continuous. Thus, we really need connectedness. – erz May 09 '17 at 00:58
  • See the scan https://math.stackexchange.com/a/1397290/4280 as well. – Henno Brandsma May 09 '17 at 06:56