Finite Product Space is Connected iff Factors are Connected/Basis for the Induction

Theorem
$\newcommand{\pr}{\operatorname {pr}}$ Let $$T_1$$ and $$T_2$$ be topological spaces.

Then the product space $$T_1 \times T_2$$ is connected iff $$T_1$$ and $$T_2$$ are connected.

Proof

 * If $$T_1 \times T_2$$ is connected then, by Projection from Product Topology is Continuous, $$T_1$$ and $$T_2$$ are continuous images under the projections $$\pr_1$$ and $$\pr_2$$.

Hence by Continuous Image of Connected Space is Connected, $$T_1$$ and $$T_2$$ are connected.


 * Suppose that $$T_1$$ and $$T_2$$ are connected.

Let:
 * $$I = T_2$$;
 * $$\forall y \in T_2: C_y = T_1 \times \left\{{y}\right\}$$;
 * $$B = \left\{{x_0}\right\} \times T_2$$ for some fixed $$x_0 \in T_1$$.

Each $$C_y$$ is homeomorphic to $$T_1$$ by Topological Product with Singleton.

By Connectedness is a Topological Property, each $$C_y$$ is therefore connected.

By the same argument, $$B$$ is also connected.

Also:
 * $$C_y \cap B = \left\{{\left({x_0, y}\right)}\right\}$$ and hence is non-empty;
 * $$\displaystyle T_1 \times T_2 = B \cup \bigcup_{y \in T_2} C_y$$.

So by the corollary to Spaces with Connected Intersection have Connected Union, it follows that $$T_1 \times T_2$$ is connected.