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Prove : A topological space $X$ is connected if and only if the only continuous functions form $X$ into the discrete space $Y={0,1}$ are the constant functions, $f(x)=0$ or $f(x)=1$.

Bérénice
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YEON
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1 Answers1

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Suppose $X$ is connected. Let $f$ be a continuous function from $X$ to $\{ 0,1\}$. Then $f(X)$ is a connected subset of $\{ 0,1\}$. The connected subsets of $\{ 0,1\}$ are $\{ 0\}$ and $\{ 1\}$ (empty set is not possible,functions have to take some value),hence $f$ must be a constant function.

Suppose the only continuous functions from $X$ to $\{ 0,1\}$ are constant. To show that $X$ is connected, we will assume it is not, and derive a contradiction. Suppose $X=A \cup B$, where $A$ and $B$ are disjoint non-empty open sets. Consider the function: $$ f(x)=\begin{cases} 0 \quad \text{if $x \in A$} \\ 1 \quad \text{if $x \in B$} \end{cases} $$ Note that $f$ is continuous ($f^{-1}$ of any subset of $\{ 0,1\}$ is an open set), and is not constant. This gives a contradiction. Hence $X$ must be connected.

This is the standard way to approach a problem like this. Please ask if any doubts.

Sarvesh Ravichandran Iyer
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    The empty topological space is connected. However there are no nonconstant functions from the empty set, so one can safely assume $X$ is not empty. – egreg Apr 13 '16 at 10:52
  • @egreg I've found there is some debate over the topic, it's a matter of taste. Look at the wikipedia page for connectedness, for example. But all said and done, what you say is correct, I'll edit my post. – Sarvesh Ravichandran Iyer Apr 13 '16 at 10:59