Dear George, There is indeed such a connection between asymmetric distances and directed algebraic topology, but is not entirely trivial. The obvious solution, by 'left' and 'right topologies' would not work well. The good solution, in my opinion, is constructing a 'symmetric topology' and adding distinguished paths. All this is in my book, and also in a paper: M. Grandis, The fundamental weighted category of a weighted space: From directed to weighted algebraic topology, Homology Homotopy Appl. 9 (2007), 221-256. I am presently working with a colleague. Later, I will be able to comment more precisely on these points. All the best Marco On 28 Sep 2009, at 20:43, George Janelidze wrote:
Dear Colleagues,
In addition to my message of September 22 addressed to Michael Barr and all of you (concerning 'Directed Algebraic Topology'/quasi-uniform spaces) I am forwarding a message from Guillaume Brummer:
"...Thank you for copying this interesting material to me, and for mentioning quasi-uniform spaces to Michael Barr. Serious early work in this field was by Leopoldo Nachbin of Rio de Janeiro, published in CRASP 226 (1948) 774-775, -- just 11 years after Andre' Weil's monograph on uniform spaces. Then came Nachbin's monograph Topologia e ordem (Chicago 1950), of which an English translation Topology and order was published by Van Nostrand (1964). Meantime the book by A'. Csa'sza'r, Fondements de la topologie ge'ne'rale, had appeared in Paris (1960)..."
Guillaume Brummer also says:
"...Hans-Peter K"unzi has published several surveys of this field since 1993, with lots of bibliography..."
However, I still see no connection (which does not mean that it will never occur!) between "directed/asymmetric algebraic topology" and "asymmetric general topology". Surely Marco Grandis is the right person to ask about this. Well, since Marco mentioned preordered topological spaces, one could think of bitopological spaces and then quasi-uniform spaces might occur naturally...
George Janelidze
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