Proposition 3.2.3.label Under each of the following conditions $A\suf X$ is connected:
- (1)
$A=\cl{Z}$ for some connected $Z$.
- (2)
$A=\cups{}{i\in I}A_{i}$ where $A_{i}$ is connected for each $i\in I$ and $\caps{}{i\in I}A_{i}\neq\emp$.
Proof. Let $f\in C(A;\curl{0,1})$ then by Definition 3.1.10
- (1)
Observe $f|_{Z}$ is continuous hence $f\equiv c\in \curl{0,1}$ so $f\inv(c)$ is closed and contains $Z$. We conclude $A=\cl{Z}\suf f\inv(c)$ showing $f$ is constant.
- (2)
Let $p\in \caps{}{i\in I }A_{i}$ then for each $i\in I$, $f|_{A_i}\equiv f(p)$ hence $f(A)=f(p)$.
$\square$
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