Completeness in topological and algebraic structures

In this project we plan to investigate completeness in different mathematical structures. Despite there are numerous different notions of completeness, most of them share a similar categorical property. Namely, a mathematical object is complete if it is closed (in some sense) in all other mathematical objects with similar structure. In this project we investigate completeness based on topological closure. Topological spaces possessing completeness properties of this kind behave similar to compact spaces, who form a well-studied important segment of general topology. Simple examples of compact spaces are finite topological spaces and the unit interval endowed with the usual topology. An interesting feature of this project is that universal closure can be provided by particular algebraic properties. This motivates us to investigate completeness of sets endowed with algebraic and topological structures. It is natural to assume that these structures are compatible with each other, which in most cases means the continuity of algebraic operations with respect to the topological structure. A seminal example of this kind is the real line, which possesses an algebraic structure defined by the operations of addition and multiplication as well as a topological structure defined by the usual metric. Moreover, the operations of addition and multiplication are continuous. In this project we also plan to investigate completeness of special algebraic and topological structures which depend on axiomatic assumptions. In this particular case adding an additional consistent axiom we can change properties of the considered structure. This is the place where logic is especially vital. To sum up we plan to work with complete topological and algebraic structures using methods that involve set theory and mathematical logic. Such an interdisciplinary approach seems to be quite innovative and thus we believe that this project will allow us to deepen the understanding of completeness.