Compact-like topological spaces and groups

The main purpose of this project is to study countably compact spaces and groups in the context of different areas of mathematics: algebra, topology and logic. Intuitively speaking, countably compact spaces are mathematical spaces in which every infinite set admits a point which is very close to it. The unit interval of the real line is among the examples of such spaces. Countably compact spaces have some nice properties and usually behave like finite spaces. In particular, any real-valued function on countably compact space is bounded, i.e., it possesses minimum and maximum. A group is an algebraic notion which means a set endowed with a some kind of multiplication or addition. For example, the real line endowed with the usual addition is an example of a group. The main motivation for studying these objects are purely mathematical. Roughly speaking, they can help to understand better some objects and actions in geometry and topology. We will be interested in the interplay of topological and algebraic properties of these objects, that is, how topological structure affects algebraic and vice versa. Beside having nice properties, these objects are quite delicate and sometimes their existence can depends on some additional axioms. It sounds unexpectedly, but the existence of some countably compact groups cannot be neither proved nor disproved in ordinary mathematics. This is the place where logic starts to play. We shall use and develop topological, algebraic and logical methods. Such an interdisciplinary approach seems to be completely innovative and thus we believe that our project will allow us to deepen our understanding of countably compact spaces and groups.

In this project we investigated compact-like topological spaces and the interplay between algebraic and topological structures on compact-like topological groups and semigroups. We constructed a compact-like topological space which possesses only constant continuous functions into the real line. This result answered a few questions of Tzannes. We improved the preceding example and constructed the first known consistent example of a rigid Nyikos space. Also, we investigated subsets of the real line with special topological and combinatorial properties related to their open covers. In particular we solved an old problem of Gerlits and Nagy. This allowed us to constructed spaces of continuous functions with peculiar local properties. We characterised commutative semigroups which are unconditionally closed in ambient topological semigroups. A discrete countable group G is unconditionally closed in the class of topological semigroups whose all finite subsets are closed if and only if G possesses no compatible (with its algebraic operation) topological structure. We discovered an algebraic property which is responsible for unconditional closeness of countable cancellative semigroups. This property implies automatic continuity of inversion in paratopological groups. Also we describe all compact topologies compatible with the semigroup operation in McAlister semigroups. It is proved that each locally compact topology compatible with algebraic structure of McAlister semigroups that corresponds to positive integers is either compact or discrete. However, this is not true for McAlister semigroups which corresponds to infinite ordinals. Finally we got a combinatorial description of finite semigroups in terms of their chains and antichains. The methods of this project were of set-theoretic nature. In particular, some of the counterexamples constructed in this project depend on axiomatic assumptions. This amplifies interdisciplinarity of the project.