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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeFeb 15th 2025
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## Idea An intrinsic notion of an [[open subobject]] in an [[elementary topos]]. ## Definition A monomorphism $U\to X$ in an [[elementary topos]] $E$ is a __Penon open__ if the following statement holds in the [[internal logic]] of $E$: $$\forall x\in X \forall y\in X(x\in U)\to (\neg(y=x)\vee y\in U).$$ ## Properties If $U\to X$ is a Penon open, then $$\forall x \in U(\{y\in X\mid \neg\neg(y=x)\}\subset U).$$ ## Related concepts * [[open subspace]] ## References * [[Jacques Penon]], _De l'infinitésimal au local_ (Thèse de Doctorat d'État), Diagrammes S13 (1985), 1-191. [numdam](http://www.numdam.org/item/DIA_1985__S13__1_0/). * [[Eduardo J. Dubuc]], [[Jacques Penon]], _Objets compacts dans les topos_, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 40:2 (1986), 203-217. [doi](https://doi.org/10.1017/s144678870002718x). * [[Jacques Penon]], _Infinitésimaux et intuitionnisme_, [[Cahiers de topologie et géométrie différentielle]] 22:1 (1981), 67-72. [numdam](http://www.numdam.org/item/CTGDC_1981__22_1_67_0/). * Oscar P. Bruno, _Logical opens of exponential objects_, [[Cahiers de Topologie et Géométrie Différentielle Catégoriques]] 26:3 (1985), 311-323. * [[Marta C. Bunge]], Felipe Gago, Ana María San Luis, [[Synthetic Differential Topology]], Cambridge University Press, 2018. ISBN: 9781108553490, [DOI](https://doi.org/10.1017/9781108553490). <a href="https://ncatlab.org/nlab/revision/Penon+open/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Penon+open">current</a>

   Created:

   Idea

   An intrinsic notion of an open subobject in an elementary topos.

   Definition

   A monomorphism in an elementary topos is a Penon open if the following statement holds in the internal logic of :

   Properties

   If is a Penon open, then

   Related concepts

   • open subspace

   References

   • Jacques Penon, De l’infinitésimal au local (Thèse de Doctorat d’État), Diagrammes S13 (1985), 1-191. numdam.

   • Eduardo J. Dubuc, Jacques Penon, Objets compacts dans les topos, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 40:2 (1986), 203-217. doi.

   • Jacques Penon, Infinitésimaux et intuitionnisme, Cahiers de topologie et géométrie différentielle 22:1 (1981), 67-72. numdam.

   • Oscar P. Bruno, Logical opens of exponential objects, Cahiers de Topologie et Géométrie Différentielle Catégoriques 26:3 (1985), 311-323.

   • Marta C. Bunge, Felipe Gago, Ana María San Luis, Synthetic Differential Topology, Cambridge University Press, 2018. ISBN: 9781108553490, DOI.

   v1, current