Definition 3.2.1 (Connected).label Let $X$ be a topological space, the following are equivalent:
- (C1)
Any open covering of $X$ by 2 nonempty sets is not disjoint as in Definition 3.1.4.
- (C2)
Any closed covering of $X$ by 2 nonempty sets is not disjoint as in Definition 3.1.4.
- (C3)
Endow $\curl{0,1}$ with discrete topology, $C(X;\curl{0,1})$ is the set of constant functions $X\to \curl{0,1}$.
- (C4)
Any clopen subset $U\suf X$ satisfies at least one of:
- (a)
$U=\emp$
- (b)
$U=X$
- (C5)
Any open covering of $X$ by at least 2 nonempty sets admits a pair of nondisjoint sets.
If the above holds then $X$ is connected.
Proof. Observe any disjoint open (resp. closed) covering of a topological space by 2 sets is also a closed (resp. open) covering hence (C1) is equivalent to (C2)
Suppose (C2) holds and $f:X\to \curl{0,1}$ is surjective then $\curl{f\inv(\curl{0}),f\inv(\curl{1})}$ is disjoint cover of $X$ by 2 sets so $f\nin C(X;\curl{0,1})$ which shows (C3).
Suppose (C3) holds. Let $U\suf X$ is clopen then the characteristic function
is continuous hence constant showing (C4).
Suppose (C4) holds. If $\curl{U_i}_{i\in I}$ is a disjoint open cover of $X$ by nonempty sets then fix $j\in I$. Observe $X\del U_{j}=\cups{}{j\neq i\in I }U_{i},U_{j}$ are open so by (C4) $U_{j}=X$ then for any $i\in I$ we have $U_{i},U_{j}$ is the desired pair of nondisjoint sets.
By hypothesis (C5) implies (C1).$\square$
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