Knowledge spaces from a topological point of view

In this paper we consider the operations of restriction, extension and gluing of pre-topologies, antimatroids, neighborhood systems and heritage choice functions. Pre-topologies are also known as knowledge spaces in knowledge space theory, and some of these operations were recently discussed in a paper by Falmagne [Falmagne, J.-C. (2008). Projections and symmetric expansions of a learning space. arXiv:0803.0575]. We introduce the notion of a separator, which allows us to decompose such objects into simpler parts, or conversely, to assemble them from simpler parts.