From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10899 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: Steven Vickers Newsgroups: gmane.science.mathematics.categories Subject: Terminology for point-free topology? Date: Mon, 16 Jan 2023 11:50:34 +0000 Message-ID: Reply-To: Steven Vickers Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="14706"; mail-complaints-to="usenet@ciao.gmane.io" To: "categories@mta.ca" Original-X-From: majordomo@rr.mta.ca Mon Jan 16 22:02:52 2023 Return-path: Envelope-to: gsmc-categories@m.gmane-mx.org Original-Received: from smtp2.mta.ca ([198.164.44.75]) by ciao.gmane.io with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.92) (envelope-from ) id 1pHWd1-0003dF-QO for gsmc-categories@m.gmane-mx.org; Mon, 16 Jan 2023 22:02:51 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:38960) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1pHWc9-00015E-W1; Mon, 16 Jan 2023 17:01:57 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1pHWbW-0003un-VX for categories-list@rr.mta.ca; Mon, 16 Jan 2023 17:01:19 -0400 Accept-Language: en-US Content-Language: en-US Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10899 Archived-At: I'm wondering if there's any consensus usage to found for "point-free" topo= logy and related terms. I've posted a detailed discussion on https://arxiv.org/abs/2206.01113, but = I can summarize the question more succinctly. It's not unusual to distinguish between two synonymous pairs: point-set/pointwise =3D ordinary semantics of general topology, point-free/pointless =3D reformed semantics of, e.g., locales or formal t= opology. However, that is misleading, as locale theory can be validly done using poi= nts. See, e.g., Ng-Vickers on real exp and log, https://lmcs.episciences.or= g/9879. The trick is to restrict to geometric constructions and to apply th= em to *generalized* points, to be found in arbitrary Grothendieck toposes a= nd not just Set (or your chosen base S). Thus there are two distinctions to be made - 1 Ordinary semantics v. reformed 2 Use points v. avoid them Some terms naturally fall into place. Point-set =3D ordinary topology, points taken from a given set. Pointwise =3D use points. Point-set is a subclass of pointwise, but strict,= as shown by the above example. What about pointless and point-free? I'm piloting - Pointless =3D avoid points (e.g. construct locale maps concretely as frame = homomorphisms). There's some value judgement in my choice there, as very of= ten the pointwise reasoning is simpler and more transparent, so there seems= to be no good reason for arguing pointlessly. Point-free =3D reformed topology. I try to think of this as meaning that th= e points are liberated from their confinement to Set or S. Does anyone have comments on these, or suggestions for other phrases for th= e concepts? Happy New Year! Steve Vickers. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]