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

We all know the classical proof that a path-connected topological space $X$ is also connected. I will recall it briefly, so that we all talk about the same thing.

Let $X$ be a path-connected topological space and assume for a contradiction that $X=A \cup B$, where $A \cap B = \emptyset$ and $A,B \neq \emptyset$ (i.e. $X$ isn't connected). Choose $a \in A$, $b \in B$ Let $\gamma: [0,1] \rightarrow X$ be a path in $X$ such that $\gamma(0)=a$, $\gamma(1)=b$. Then by continuity of $\gamma$ there is a decomposition $$[0,1]=\gamma^{-1}(A) \cup \gamma^{-1}(B)$$ and this decomposition implies that $[0,1]$ is not connected, which is not true. Hence we conclude that $X$ cannot be path-connected, if it isn't connected.

Is there a proof, which does not use the connectedness of $[0,1]$? I would like a slick short proof similar to the one above, if you know one. Or is there a good reason, why the connectedness of $[0,1]$ should be essential?

• Why do you know that $[0,1]=\gamma^{-1}(X)?$
– user
Commented May 28, 2015 at 4:44

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.

• Thank you very much for this enlightening answer! Commented Apr 4, 2012 at 8:54

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.

• Unfortunately, this uses the fact that [0,1] is connected: "Each py ([0, 1]) is connected since the image of a connected set under a continuous function is connected." This proof does not answer the author's question. Commented Mar 26, 2023 at 22:07