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abstract: 'In a projective space we fix some set of points, a horizon, and investigate the complement of that horizon. We prove, under some assumptions on the size of lines, that the ambient projective space, together with its horizon, both can be recovered in that complement. Then we apply this result to show something similar for Grassmann spaces.'
author:
- 'K. Petelczyc and M. Żynel'
title: The complement of a point subset in a projective space and a Grassmann space
---

Mathematics Subject Classification (2010): 51A15, 51A45.\
Keywords: projective space, affine space, Grassmann space, slit space, complement.

Introduction
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Our goal
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Two definitions of an affine space are known: it is a projective space with one of its hyperplanes removed, or it is a projective space with one of its hyperplanes distinguished. It is also known that these two definitions are equivalent, since the removed hyperplane can be recovered in terms of so obtained affine space. A general question arises: how big the remaining fragment of a projective space must be so as the surrounding space can be recovered in terms of the internal geometry of this fragment? Analogous question could be asked in the context of projective 