Equivalence of Definitions of Connected Topological Space

Theorem
Let $T$ be a topological space.

Then the following definitions of connectedness:


 * $(1): \quad T$ is connected iff there does not exist any continuous surjection from $T$ onto a discrete two-point space


 * $(2): \quad T$ is connected iff it admits no partition

are equivalent.

Proof
Let $D$ be the set $\left\{{0, 1}\right\}$ with the discrete topology.


 * Suppose $A | B$ partitions $T$.

Let us define $f: T \to D$ by:

$f \left({x}\right) = \begin{cases} 0 & : x \in A \\ 1 & : x \in B \end{cases}$

$A | B$ is a partition so neither of $A$ and $B$ is empty.

Hence $f$ is a surjection.

As $D$ is a discrete space, the open sets of $D$ are the elements of $\mathcal P \left({D}\right)$ where $\mathcal P \left({D}\right)$ is the power set of $D$.

Here, then, $\mathcal P \left({D}\right) = \left\{{\varnothing, \left\{{0}\right\}, \left\{{1}\right\}, D}\right\}$.

Thus:

Thus $f$ is continuous, since each of $\varnothing, \left\{{0}\right\}, \left\{{1}\right\}, D$ is open in $D$.

Hence, by definition $(1)$, $T$ is not connected.


 * Suppose there exists a continuous surjection $f: T \to D$.

Then the fact that $f^{-1} \left({\left\{{0}\right\}}\right) | f^{-1} \left({\left\{{1}\right\}}\right)$ partitions $T$ is immediate.