$C_p(\omega_1)$ is a Lindelöf space

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From time to time I come across the following statement:

$C_p(\omega_1)$ is a Lindelöf space.

Where can I find a proof of this?

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I finally found it.

Problem 316 (and its solution) in Tkachuk's "A Cp-theory problem book" (2011), page 280.