TITLE: Product of path connected spaces is path connected
QUESTION [13 upvotes]: Will product of path connected topological spaces be necessarily path connected? Why or why not?
Give me some hints. Thank you.

REPLY [33 votes]: This is very straightforward from applying the definitions involved:
Assume $X=\prod_{i\in I}X_i$ with $X_i$ path-connected.
Let $x=(x_i)_{i\in I}$, $y=(y_i)_{i\in I}$ be two points in $X$.
By assumption, thee exist continuous paths $\gamma_i\colon[0,1]\to X_i$ with $\gamma_i(0)=x_i$ and $\gamma_i(1)=y_i$.
By definition of product, there exists a unique continuous $\gamma\colon[0,1]\to X$ such that $\pi_i\circ\gamma=\gamma_i$ for all $i\in I$ where $\pi_i$ is the projection $X\to X_i$. That makes $\gamma$ a path from $x$ to $y$.