# Projection from Product Topology is Continuous

## Theorem

Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.

Let $T = \struct {T_1 \times T_2, \tau}$ be the product space of $T_1$ and $T_2$, where $\tau$ is the product topology on $S$.

Let $\pr_1: T \to T_1$ and $\pr_2: T \to T_2$ be the first and second projections from $T$ onto its factors.

Then both $\pr_1$ and $\pr_2$ are are continuous.

### General Result

Let $\family {T_i}_{i \mathop \in I} = \family {\struct{S_i, \tau_i}}_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.

Let $\ds S = \prod_{i \mathop \in I} S_i$ be the corresponding product space.

Let $\tau$ denote the Tychonoff topology on $S$.

Let $\pr_i: S \to S_i$ be the corresponding projection from $S$ onto $S_i$.

Then $\pr_i$ is continuous for all $i \in I$.

## Proof

From Natural Basis of Product Topology of Finite Product, a basis for $\tau$ is:

- $\BB = \set {U \times V: U \in \tau_1, V \in \tau_2}$

Let $U$ be open in $T_1$.

Then $\map {\pr_1^{-1} } U = U \times T_2$ is one of the open sets in the basis in the definition of product topology.

Thus $\pr_1$ is continuous.

The same argument can be applied to $\pr_2$.

$\blacksquare$

## Also see

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.5$: Products: Proposition $3.5.3$