Showing a Set is Path Connected

Consider the set $X$ such that $X = [0,1] \times \{0\} \cup \bigcup_{q\in \mathbb{Q}\cap [0,1]} \{q\} \times [0,1] $

I know that from definition, a set is path-connected if $\forall x,y \in X$, $\exists$ a continuous map $\gamma:[0,1] \to X$ such that $\gamma(0)=x \text{ and } \gamma(1)=y$. If I could get a hint on how to approach this I would appreciate it.

Solution 1:

If there is a path from $x$ to $y$ in a set $A$, and also one from $y$ to $z$, show that there is also a path in $A$ from $x$ to $z$.

Using that show that if $A=\bigcup_{i\in I} A_i$ is a union of path connected sets $A_i$ with the additional property that for some fixed index $j \in I$ we have $A_j \cap A_i \neq \emptyset$ for all $i$, then $A$ is path-connected.

The latter applies to your problem quite directly.