In this talk, we will consider special sets of reals that arise naturally in real analysis, general topology, and algebra, yet
can be defined using elementary set-theoretic operations on the reals. These sets exhibit a surprisingly rich infinitary combinatorial
structure, and their study often brings us to the boundaries of our axiomatic systems. Typical examples include maximal almost disjoint families, maximal independent families, bounded and unbounded families, and maximal cofinitary groups. We will discuss recent advances and emerging trends in their study, and outline interesting remaining open questions.
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