Theorem 3.5:

(i) If $X$ and $Y$ are topological spaces and $f : X \longrightarrow Y$ is a continuous map and $A$ is a path connected subset of $X$ then $f(A)$ is also a path connected subset of $Y$.

(ii) If $\{A_n\}$ is a sequence of path connected subsets of a topological space $X$ and $A_n\cap A_{n+1}$ is non-empty for each $n = 1, 2, 3, \dots$ then $\cup_{n=1}^{\infty}A_n$ is path connected. In particular, taking $A_2 = A_3 = \dots$ we see get the result for two path connected sets.