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 $5.6.3$