General Topology – Continuous Function from (X,?) to {0,1} with Discrete Topology

connectednessgeneral-topology

Let $f$ be a continuous function such that $f : (X,\tau) \rightarrow (\{0,1\},\tau_1\}$. Where $(X,\tau)$ is a generic topological space and $\tau_1$ is the discrete topology. I want to prove that if f is non-constant then $(X,\tau)$ is disconnected.

I started by describing $(\{0,1\},\tau_1\}$. This topological space is compact, totally disconnected and Hausdorff. However,from here I do not know how to continue. Any tips?

Best Answer

Hints:

  1. An image of a connected set via continuous mapping is connected.
  2. A nonempty subset of a discrete space is connected if and only if it is a singleton (i.e. contains exactly one point).
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