# Topological Product of Compact Spaces/Finite Product

## Theorem

Let $T_1, T_2, \ldots, T_n$ be topological spaces.

Let $\ds \prod_{i \mathop = 1}^n T_i$ be the product space of $T_1, T_2, \ldots, T_n$.

Then $\ds \prod_{i \mathop = 1}^n T_i$ is compact if and only if all of $T_1, T_2, \ldots, T_n$ are compact.

## Proof

Proof by induction:

For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:

- $\ds \prod_{i \mathop = 1}^n T_i$ is compact if and only if all of $T_1, T_2, \ldots, T_n$ are compact

### Basis for the Induction

$\map P 1$ is the case:

- $T_1$ is compact if and only if $T_1$ is compact

which is trivially true.

Thus $\map P 1$ is seen to hold.

This is the basis for the induction.

### Induction Hypothesis

Now it needs to be shown that if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.

So this is the induction hypothesis:

- $\ds \prod_{i \mathop = 1}^k T_i$ is compact if and only if all of $T_1, T_2, \ldots, T_k$ are compact

from which it is to be shown that:

- $\ds \prod_{i \mathop = 1}^{k + 1} T_i$ is compact if and only if all of $T_1, T_2, \ldots, T_{k + 1}$ are compact

### Induction Step

This is the induction step:

We have:

- $\ds \prod_{i \mathop = 1}^{k + 1} T_i = \paren {\prod_{i \mathop = 1}^k T_i} \times T_{k + 1}$

Hence:

\(\ds \prod_{i \mathop = 1}^{k + 1} T_i\) | \(\text {is}\) | \(\ds \text {compact}\) | ||||||||||||

\(\ds \leadstoandfrom \ \ \) | \(\ds \prod_{i \mathop = 1}^k T_i, T_{k + 1}\) | \(\text {are}\) | \(\ds \text {compact}\) | Topological Product of Compact Spaces | ||||||||||

\(\ds \leadstoandfrom \ \ \) | \(\ds T_1, T_2, \ldots, T_k, T_{k + 1}\) | \(\text {are}\) | \(\ds \text {compact}\) | Induction Hypothesis |

So $\map P k \implies \map P {k + 1}$ and thus it follows by the Principle of Mathematical Induction that for any $n \ge 1$:

- $\ds \prod_{i \mathop = 1}^n T_i$ is compact if and only if all of $T_1, T_2, \ldots, T_n$ are compact

$\blacksquare$

## Also see

- Tychonoff's Theorem, where this result is extended to the topological product of any infinite number of topological spaces.

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $5$: Compact spaces: $5.6$: Compactness and Constructions: Remark $5.6.3$