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    $\begingroup$ this is just the same as kuratowski's closure operator. It is also confusing because there are also the Nearness spaces defined by Herrlich which use covers and are more general than topological spaces.. $\endgroup$ Commented Mar 24, 2010 at 3:53
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    $\begingroup$ @jef: Yes, it's the same as the closure operator, but I find it way easier to motivate. Maybe I'm just weird? Thanks for mentioning Herrlich's nearness spaces---I'd never heard of them before! If anyone is interested in the definition, here's a nice reference: "On Nearness Space," by Seung On Lee and Eun Ai Choi (ccms.or.kr/data/pdfpaper/jcms8_1/8_1_19.pdf). $\endgroup$ Commented Mar 24, 2010 at 22:51
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    $\begingroup$ @Vectornaut: Your definition of a continuous map is a nice formalization of informal “a map without breaks” or “a map which preserves infinitesimal distances”. a touches A ↔ a is infinitely close to A. Infinitesimal distance between points does not make sense, so we need to replace one of the points with a set. That was crucial. I went the same way, but directly from the Kuratowski's axioms. a touches A ↔ $a\in cl(A)$. Thank you very much for the references because I did not know even where to start my search or what name it is called. I wonder why point-set topology is not derived this way. $\endgroup$ Commented Feb 19, 2011 at 14:37
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    $\begingroup$ Having studied various non-standard topologies, especially ones relating to adeles and profinite groups, I find this axiomatization very useful. $\endgroup$ Commented May 9, 2012 at 9:32
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    $\begingroup$ @JonaChristopherSahnwaldt, it's not silly at all! Using $\in$ as the "touches" relation gives the discrete topology. $\endgroup$ Commented Jul 17, 2016 at 18:41