We present some properties recently introduced by Bal and Bhowmik: the $R$-, $H$-, and $M$-star Lindelöfness. These properties are defined via selection principles involving the star operator and lie between covering properties (in particular, star-covering properties) and certain selective strengthenings of separability. This dual perspective leads to several implications and connections among known properties, some of which we present. Additionally, we provide some examples. In particular, we prove that there exists a Tychonoff M-star Lindelöf space of cardinality $\frak{c}$ which is not $R$-star Lindelöf answering a question posed by Bal and Bhowmik.
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Selection principles and their corresponding games involving points and open covers have a long and well-developed history. In this talk, we will highlight how many results for points and open covers have analogues involving compact sets and k-covers. Such examples will include traditional Menger/Rothberger variants, as well as various connections with the space of real-valued continuous functions with the topology of uniform convergence on compacta. Most of the results to be discussed come from joint work with Steven Clontz and Jared Holshouser.
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Saharon Shelah proved a remarkable theorem of cardinal arithmetic in 1989: if $\aleph_\omega$ is a strong limit cardinal, then $2^{\aleph_\omega}<\aleph_{\omega_4}$. His proof used the tools of pcf theory (a body of set-theoretic tools that are helpful in analyzing certain types of infinite products) but it was soon realized that once the basic ingredients of pcf theory are given, the rest of the argument is essentially topological. Our aim is to survey the topological aspects Shelah’s proof, and present some recent applications.
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