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Tag Archives: filters
When do products distribute over colimits in Top?
In a recent paper, Lawson and Xu give a new class of posets on which the Scott topology of the (poset) product of two posets coincides with the product topology (of each poset with its Scott topology). I will explain … Continue reading
Posted in Uncategorized
Tagged colimit, core-compactness, counterexample, filters, product
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Hoover’s maximal limit spaces II: products, liftings, retracts, function spaces, and hyperspaces
Last time, we had introduced Hoover’s maximal limit spaces: spaces in which every convergent filter has a unique largest limit. That notion is closed under many constructions, as we will see: products, liftings, retracts, notably, and that is elementary. The … Continue reading
Posted in Uncategorized
Tagged coherence, convergence, filters, function space, hyperspace, sobriety, strong sobriety, weak Hausdorffness
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Hoover’s maximal limit spaces I: local strong sobriety, bounded sup-completeness and weak Hausdorffness
In 1995, Douglas Hoover introduced and studied a notion of maximal limit spaces: spaces in which every convergent net has a unique largest limit. This has connections with many other kinds of spaces that we have explored already, and I … Continue reading
Posted in Uncategorized
Tagged coherence, convergence, filters, sobriety, strong sobriety, weak Hausdorffness
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Weakly Hausdorff spaces, and locally strongly sober spaces
A funny convergence of topics happened a few weeks ago. Frédéric Mynard told me about so-called locally strongly sober spaces (which, I am ashamed to say, I had heard about but completely forgotten about). At the same time, I was … Continue reading
Posted in Uncategorized
Tagged coherence, convergence, filters, sobriety, strong sobriety, weak Hausdorffness
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Convergence without points
Can you define convergence without mentioning points? More precisely, is there any form of Stone duality for convergence spaces, instead of just topological spaces? The short answer is yes. For the long answer, read the full post.
Posted in Uncategorized
Tagged convergence, duality, filters, Stone duality
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The Rasiowa-Sikorski lemma and the Baire property
Merry Christmas! And a Happy New Year, too. There are incredible links between logic and topology, and I would like to start with something called the Rasiowa-Sikorski lemma in logic. That is a theorem that states the existence of certain … Continue reading
Posted in Uncategorized
Tagged Baire property, filters, logic
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