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Is there a book, which expresses all the questions, or searches for attainment of certain utility/need, or thing, which gave the discovery or invention of all the components of Topology?

I need book to give more math data; I want to know topology from that, than from a non-historical ordered textbook, as abstraction if has come from certain concrete contexts, they may allow knowing on how entire structure of topology is made for those contexts, and later allow making their alternate structures for our other needs. This might allow having memory of topology, from their particular needs which have created them, than having memory from the continuous artificial reknowing?

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I. M. James, History of Topology.

This is a collection of 40 essays by different authors, on topics related mostly to manifolds and algebraic topology.

Perhaps the page

http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Topology_in_mathematics.html

will be of interest for aspects of topology before Poincaré came on the scene.

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I don't entirely understand the question, but this book is probably relevant:

J. H. Manheim, The Genesis of Point Set Topology (1964).

I must have borrowed it from a library a long time ago. It is out of print. Used copies go for about a hundred quid. I suppose Dover might look kindly on a request for a reprint.

This is very probably also relevant (and it is certainly a very good read):

José Ferreirós, Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics (1999; second edition Birkhäuser, Basel 2007).

It is expensive, but perhaps it will appear again in one of Springer's annual Yellow Sales.

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J. Dieudonne, A History of Algebraic and Differential Topology, 1900 - 1960.

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