Problem 2

Exercise 9.2 (Continuity and connectedness in discrete topologies)   Let $X$ be a topological space and $\mathbb{Z}$ in the standard topology. Consider the property

$$P(X) := f : X\to\mathbb{Z}$$ (9.11)

1. With $\mathbb{R}$ in the standard topology, show that $P(\mathbb{R})$ is true.
2. For an arbitrary topological space $X$, find and prove a characterization of $P(X)$ in terms of $X$.

Proof. With $\mathbb{R}$ in the standard topology, we $P(\mathbb{R})$ is true. This can be seen by making a metric space argument. Let $R>0$ be fixed and a select $\delta>0$ so that $\vert f(x)-f(y)\vert<1/2$ for any $x,y\in[-R,R]$. Each integer is isolated, so this inequality implies $f(x)=f(y)$, so that $f$ is constant on expanding intervals, indicating $f$ is constant on $\mathbb{R}$.

Now we show $P(X)$ is true if and only if $X$ is connected.

Suppose $X$ is connected. The continuous image of a connected space is connected, so $f(X)$ is connected. The connected subsets of $\mathbb{Z}$ are precisely the singletons, so we know $f(X)=\{n\}$, indicating $f$ is constant.

Conversely, suppose $X$ is not connected. It is possible to define a continuous function which is non-constant by separating $X=A\cup B$ and defining $f(A)=1$ and $f(B)=0$. The open sets in $\mathbb{Z}$ are precisely the singletons, so any preimage equals $A$, $B$, or is trivial, so that $f$ is continuous. Therefore if $X$ is connected, every continuous function $f : X\to\mathbb{Z}$ is constant. $\qedsymbol$