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adjunction space

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1. I have seen in some books this definition with the assumption that A is closed subspace. (Walker: Stone-Cech compactification, p.267; Engelking: General Topology)
I wasn't able to find a reference, where this assumption is relaxed. (But I would say, whenever I've used this notion, the condition of closedness wasn't necessary.)
Do you know about a book where adjunction space is used without this assumption?

2. Perhaps it could be mentioned in the article, that the adjunction space is pushout in the category of topological spaces.

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