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In the thesis "Topology of Function Spaces" by Andrew Marsh (Galway, 2000) it is claimed (page 23, before Definition 30) that the space $C(\omega_1)$ is Lindelöf with respect to the compact open topology. For a proof the reader is refered to

S.P. Gul'ko. On properties of $\sigma$-products. Soviet Math. Dokl. 18, 6 1977.

I searched a while on the net but I could not find another reference to this paper - it seems there is a paper by Gul'ko named "On properties of subsets of $\Sigma$-products" but I do not know wether that is the same paper or how to obtain access to the latter paper.

I wanted to ask Andrew Marsh but I could not find a way to address him (plus there seem to be more people of that name).

Furthermore, I could not find any other source for that claim.

So, my question is:

• Can you give me a reference (or a proof if it fits on one page) for the fact that $C_k(\omega_1)$ is Lindelöf? It does not need to be the Gul'ko paper above but if you manage to find it and it really contains the proof, that is of course also very fine.

• Bonus question (not really part of the question but it would be even nicer): Is it known if $C_k(\omega_1) $ is $k$-Lindelöf in the sense of my last question (see Lindelöf Property for Open Covers for Compact Sets), i.e. does every open $k$-cover admit a countable $k$-subcover? (a $k$-cover is a cover such that every compact subset is contained in one of the members of the cover).

In the thesis "Topology of Function Spaces" by Andrew Marsh (Galway, 2000) it is claimed (page 23, before Definition 30) that the space $C(\omega_1)$ is Lindelöf with respect to the compact open topology. For a proof the reader is refered to

S.P. Gul'ko. On properties of $\sigma$-products. Soviet Math. Dokl. 18, 6 1977.

I searched a while on the net but I could not find another reference to this paper - it seems there is a paper by Gul'ko named "On properties of subsets of $\Sigma$-products" but I do not know wether that is the same paper or how to obtain access to the latter paper.

I wanted to ask Andrew Marsh but I could not find a way to address him (plus there seem to be more people of that name).

Furthermore, I could not find any other source for that claim.

So, my question is:

• Can you give me a reference (or a proof if it fits on one page) for the fact that $C_k(\omega_1)$ is Lindelöf? It does not need to be the Gul'ko paper above but if you manage to find it and it really contains the proof, that is of course also very fine.

• Bonus question (not really part of the question but it would be even nicer): Is it known if $C_k(\omega_1) $ is $k$-Lindelöf in the sense of my last question (see Lindelöf Property for Open Covers for Compact Sets), i.e. does every open $k$-cover admit a countable $k$-subcover? (a $k$-cover is a cover such that every compact subset is contained in one of the members of the cover).

In the thesis "Topology of Function Spaces" by Andrew Marsh (Galway, 2000) it is claimed (page 23, before Definition 30) that the space $C(\omega_1)$ is Lindelöf with respect to the compact open topology. For a proof the reader is refered to

S.P. Gul'ko. On properties of $\sigma$-products. Soviet Math. Dokl. 18, 6 1977.

I searched a while on the net but I could not find another reference to this paper - it seems there is a paper by Gul'ko named "On properties of subsets of $\Sigma$-products" but I do not know wether that is the same paper or how to obtain access to the latter paper.

I wanted to ask Andrew Marsh but I could not find a way to address him (plus there seem to be more people of that name).

Furthermore, I could not find any other source for that claim.

So, my question is:

• Can you give me a reference (or a proof if it fits on one page) for the fact that $C_k(\omega_1)$ is Lindelöf? It does not need to be the Gul'ko paper above but if you manage to find it and it really contains the proof, that is of course also very fine.

• Bonus question (not really part of the question but it would be even nicer): Is it known if $C_k(\omega_1) $ is $k$-Lindelöf in the sense of my last question (see Lindelöf Property for Open Covers for Compact Sets), i.e. does every open $k$-cover admit a countable $k$-subcover? (a $k$-cover is a cover such that every compact subset is contained in one of the members of the cover).

In the thesis "Topology of Function Spaces" by Andrew Marsh (Galway, 2000) it is claimed (page 23, before Definition 30) that the space $C(\omega_1)$ is Lindelöf with respect to the compact open topology. For a proof the reader is refered to

S.P. Gul'ko. On properties of $\sigma$-products. Soviet Math. Dokl. 18, 6 1977.

I searched a while on the net but I could not find another reference to this paper - it seems there is a paper by Gul'ko named "On properties of subsets of $\Sigma$-products" but I do not know wether that is the same paper or how to obtain access to the latter paper.

I wanted to ask Andrew Marsh but I could not find a way to address him (plus there seem to be more people of that name).

Furthermore, I could not find any other source for that claim.

So, my question is:

• Can you give me a reference (or a proof if it fits on one page) for the fact that $C_k(\omega_1)$ is Lindelöf? It does not need to be the Gul'ko paper above but if you manage to find it and it really contains the proof, that is of course also very fine.

• Bonus question (not really part of the question but it would be even nicer): Is it known if $C_k(\omega_1) $ is $k$-Lindelöf in the sense of my last question (see Lindelöf Property for Open Covers for Compact Sets), i.e. does every open $k$-cover admit a countable $k$-subcover? (a $k$-cover is a cover such that every compact subset is contained in one of the members of the cover).

In the thesis "Topology of Function Spaces" by Andrew Marsh (Galway, 2000) it is claimed (page 23, before Definition 30) that the space $C(\omega_1)$ is Lindelöf with respect to the compact open topology. For a proof the reader is refered to

S.P. Gul'ko. On properties of $\sigma$-products. Soviet Math. Dokl., 18, 6 1977.

I searched a while on the net but I could not find another reference to this paper - it seems there is a paper by Gul'ko named "On properties of subsets of $\Sigma$-products" but I do not know wether that is the same paper or how to obtain access to the latter paper.

I wanted to ask Andrew Marsh but I could not find a way to address him (plus there seem to be more people of that name) ...

Furthermore, I could not find any other source for that claim...

So, my question is:

• Can you give me a reference (or a proof if it fits on one page) for the fact that $C_k(\omega_1)$ is Lindelöf? It does not need to be the Gul'ko paper above but if you manage to find it and it really contains the proof, that is of course also very fine.
• Bonus question (not really part of the question but it would be even nicer): Is it known if $C_k(\omega_1) $ is $k$-Lindelöf in the sense of my last question (see Lindelöf Property for Open Covers for Compact Sets), i.e. does every open $k$-cover admit a countable $k$-subcover? (a $k$-cover is a cover such that every compact subset is contained in one of the members of the cover)

Thank you very much!

Tom

In the thesis "Topology of Function Spaces" by Andrew Marsh (Galway, 2000) it is claimed (page 23, before Definition 30) that the space $C(\omega_1)$ is Lindelöf with respect to the compact open topology. For a proof the reader is refered to

S.P. Gul'ko. On properties of $\sigma$-products. Soviet Math. Dokl. 18, 6 1977.

I searched a while on the net but I could not find another reference to this paper - it seems there is a paper by Gul'ko named "On properties of subsets of $\Sigma$-products" but I do not know wether that is the same paper or how to obtain access to the latter paper.

I wanted to ask Andrew Marsh but I could not find a way to address him (plus there seem to be more people of that name).

Furthermore, I could not find any other source for that claim.

So, my question is:

• Can you give me a reference (or a proof if it fits on one page) for the fact that $C_k(\omega_1)$ is Lindelöf? It does not need to be the Gul'ko paper above but if you manage to find it and it really contains the proof, that is of course also very fine.

• Bonus question (not really part of the question but it would be even nicer): Is it known if $C_k(\omega_1) $ is $k$-Lindelöf in the sense of my last question (see Lindelöf Property for Open Covers for Compact Sets), i.e. does every open $k$-cover admit a countable $k$-subcover? (a $k$-cover is a cover such that every compact subset is contained in one of the members of the cover).