Equivalence of Definitions of Connected Topological Space

Theorem
Let $$T$$ be a topological space.

Then the following definitions of connectedness:


 * 1) $$T$$ is connected iff there does not exist any continuous surjection from $$T$$ onto a discrete two-point space;
 * 2) $$T$$ 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.