Path-Connected implies Connected without knowing that [0,1] is connected

The connectedness of $[0,1]$ is clearly crucial, as may be seen by considering what would happen if we used something else.

Let $K$ be any space with two distinguished points $p$ and $q$. Say that a space $X$ is $K$-connected if for any $x,y\in X$ there is a continuous map $f:K\to X$ such that $f(p)=x$ and $f(q)=y$. If $K$ is not connected, there is clearly no reason to expect $K$-connectedness of $X$ to imply connectedness of $X$. As an extreme case, let $K=\{0,1\}$ with the discrete topology: then every space is $K$-connected.

For completeness I would just like to add another proof to this which is found here (Remark 2.2 on page 2). This proof is a direct proof, but it uses some property of connected sets. We need the following result.

If $C_i$ are connected sets and $\cap_i C_i \neq \emptyset$ then $\cup_i C_i$ is connected.

Now let $X$ be a path connected set and choose $x \in X$. Since $X$ is path connected we can for all $y\in X$ find a path $p_y$ within $X$ so that $p_y(0) = x$ and $p_y(1) = y$. Now consider the paths $p_y$ for all $y\in X$. All these paths contain the common point $x$ so their intersection is nonempty. The union of all the paths is just $X$ and by the lemma above, $X$ is hence connected.