Topology: An Inquiry-Based Approach

The product of the $X_i$ is the Cartesian product ${\mathrm{\Pi }}_{i=1}^n{X}_i\text{.}$

The mapping ${\pi }_j:{\mathrm{\Pi }}_{i=1}^n{X}_i\to X_j$ defined by ${\pi }_j((x_i))=x_j$ is the projection map onto $X_j$ for $j$ from 1 to $n\text{.}$

A function $f$ mapping a topological space $Y$ to ${\mathrm{\Pi }}_{i=1}^n{X}_i$ is continuous if and only if ${\pi }_j\circ f$ is continuous for every $j$ from $1$ to $n\text{.}$

Let $(X,\tau )$ be a topological space. A subset $\mathcal{S}$ of $\tau$ is a subbasis for $\tau$ if the set $\mathcal{S}$ of all finite intersections of elements of $\mathcal{S}$ is a basis for $\tau \text{.}$

If each $X_i$ is (a) connected, (b) path connected, (c) compact, then ${\mathrm{\Pi }}_{i=1}^n{X}_i$ is (a) connected, (b) path connected, (c) compact with respect to the product topology.