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Is there a nice characterization of topological spaces with the property that the product with any paracompact space is paracompact?

All compact spaces have this property (this can be shown from the tube lemma). But somebody once gave me an example (that I cannot locate) of a non-compact space with the property. I didn't check the example carefully, so I cannot vouch for its accuracy.

If a characterization is too hard, an example of a non-compact space would also be great.

[NOTE: I don't assume Hausdorffness in my definitions of compact and paracompact, though it would be nice if the example were a Hausdorff space.]

ADDED LATER: I forgot to mention this, but a product of paracompact spaces need not be paracompact. The standard example is the Sorgenfrey line (the real line with the lower limit topology), which is paracompact, whose product with itself, the Sorgenfrey plane, is not paracompact.

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# Space whose product with paracompact space is paracompact

Is there a nice characterization of topological spaces with the property that the product with any paracompact space is paracompact?

All compact spaces have this property (this can be shown from the tube lemma). But somebody once gave me an example (that I cannot locate) of a non-compact space with the property. I didn't check the example carefully, so I cannot vouch for its accuracy.

If a characterization is too hard, an example of a non-compact space would also be great.

[NOTE: I don't assume Hausdorffness in my definitions of compact and paracompact, though it would be nice if the example were a Hausdorff space.]