# Product of Lindelöf Spaces is not always Lindelöf

## Theorem

Let $I$ be an indexing set.

Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.

Let $\ds \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$.

Let each of $\struct {S_\alpha, \tau_\alpha}$ be a Lindelöf space.

Then it is not necessarily the case that $\struct {S, \tau}$ is also Lindelöf space.

## Proof

Let $T$ be the Sorgenfrey line.

Let $T' = T \times T$ be Sorgenfrey's half-open square topology.

From Sorgenfrey Line is Lindelöf, $T$ is a Lindelöf space.

From Sorgenfrey's Half-Open Square Topology is Not Lindelöf, $T'$ is not a Lindelöf space.

Hence the result.

$\blacksquare$