Department of Mathematics
Title : On topologies on $X$ as points within $2^{\mathcal P}(X)$: \\ lattice theory meets topology
| Speaker: Dr. Aisling McCluskey Affiliation: National University of Ireland Time: 10 am Friday, 4 May, 2012 Location: 303S-561 |
Abstract
| For a non-empty set $X$, the collection $Top(X)$ of all topologies on $X$ sits inside the Boolean lattice $ {\mathcal P}{\mathcal P}(X).$ (when ordered by set-theoretic inclusion) which in turn can be naturally identified with the Stone space $ 2^{\mathcal P}(X)$. Via this identification then, $Top(X)$ naturally inherits the subspace topology from $ 2^{\mathcal P}(X)$. Extending ideas of Frink (1942), we apply lattice-theoretic methods to establish an equivalence between the topological closures of sublattices of $ 2^{\mathcal P}(X)$ and their (completely distributive) completions. We exploit this equivalence in an investigation of the topological nature of $Top(X)$. In this talk, we describe some insights gained particularly regarding the Borel complexity of $Top(X)$. |
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Programmes and Centres
- New Zealand Institute of Mathematics and its Applications (NZIMA)
- Community for Understanding and Learning in the Mathematical Sciences (CULMS)
- Centre for Mathematical Social Science (CMSS)
- Department of Computer Science
- Department of Engineering Science
- Department of Physics
- Department of Statistics
- Auckland Bioengineering Institute
