A quotient-like construction involving elementary submodels

This article is an investigation of a method of deriving a topology from a space and an elementary submodel containing it. We first define and give the basic properties of this construction, known as $X\mathrm{/}M$. In the next section, we construct some examples and analyse the topological relationship between $X$ and $X\mathrm{/}M$. In the final section, we apply $X\mathrm{/}M$ to get novel results about Lindel"{o}f spaces, giving partial answers to a question of F.D. Tall and another question of Tall and M. Scheepers.