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I was wondering if anyone is aware of any books that provide a comprehensible and complete picture of mathematical spaces? I've looked into many real analysis as well as complex for a complete list of mathematical spaces in hierarchical view representing their relations in a natural progression allowing one to understand when can you reduce one form to the other. As I mentioned most books I've looked even well-known ones such as Rubin's don't provide this view. Even Wikipedia which is awesome doesn't have the complete picture, for instance the shobolev space doesn't come up at all on wikipedia as well as in the books I've looked. If anyone knows such a book, manuscript or technical report that would be great.

Thanks in advance!

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No such book exists because there is no such object as a space in Mathematics, let alone any sort of gradation between them. The term space is a shortcut for the name of a generic object considered in a given area, so that in Topology a space means a topological space, in Linear algebra it is a vector space, in Functional analysis it may or may not be a Banach space.

So you are really asking about a book that would give a complete picture of Mathematics. Given that Mathematics is a humungous area of human knowledge, no such a book exists, however to some extent there is a proxy for this, namely the series of books by Bourbaki. (Although I am sure this is not what you had in mind.)

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  • $\begingroup$ Thank you @Tomek that makes things even more clear, I was aware of Bourbaki series but wasn't quite sure if it was appropriate. I might have to reconsider that. $\endgroup$ – Jane Dane Jun 25 at 12:53
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This might be helpful to you:

https://topology.jdabbs.com/properties

It is a database of topological spaces where you can filter by properties. It seems pretty cool; I believe it is called ''$\pi$-base''.

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  • $\begingroup$ Thanks @Erin this seems extremely useful. I wasn't aware of it. Cheers! $\endgroup$ – Jane Dane Jun 25 at 12:54

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