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# Topology - Set-Theoretic

**DMS Set Theoretic Topology Seminar**

Mar 06, 2020 02:00 PM

Parker Hall 246

Speaker:

**Professor Vladimir Tkachuk**

Title: Character and tightness are discretely reflexive in compact spaces.

Abstract: We will show that tightness, character, sequentiality, and Frechet-Urysohn property are discretely reflexive in compact spaces, i.e., if \(X\) is a compact space in which the closure of every discrete subset has a property \(P\) from the list \(tightness \leq \kappa, character\leq\kappa, sequentiality, Frechet-Urysohn property\), then \(X\) has the property \(P\).

**DMS Set Theoretic Topology Seminar**

Feb 28, 2020 02:00 PM

Parker Hall 246

Speaker: **Professor Vladimir Tkachuk**

Title: Neither sigma-compactness nor Lindelof Sigma-property is discretely reflexive under CH

Abstract: We will construct, under the Continuum Hypothesis, an example of a space X in which the closure of every discrete set is countable (and hence sigma-compact) while X is not a Lindelof Sigma-space. Therefore, countability, sigma-compactness, and Lindelof Sigma-property all fail to be discretely reflexive under CH.

**DMS Set Theoretic Topology Seminar**

Feb 21, 2020 02:00 PM

Parker Hall 246

Speaker: **Professor Vladimir Tkachuk**

Title: Lindelof property is discretely reflexive in spaces of countable tightness.

Abstract: After finishing the construction of the example of a crowded countable space which is not weakly discretely generated, we will start the proof of discrete reflexivity of the Lindelof property in spaces of countable tightness. In other words, we will prove that if \(X\) is a space of countable tightness and the closure of every discrete subspace of \(X\) is Lindelof, then \(X\) itself is Lindelof. This non-trivial theorem of Arhangel'skii is not easy to prove even for first countable spaces.

**DMS Set Theoretic Topology Seminar**

Feb 14, 2020 02:00 PM

Parker Hall 246

Speaker: **Professor Vladimir Tkachuk**

Title: A countable space need not be weakly discretely generated

Abstract: We will construct a countable crowded Tychonoff space \(X\) in which all discrete subsets are closed. This very non-trivial example of van Douwen shows that a discretely metrizable space is not necessarily first countable or even Frechet-Urysohn. This space has no non-trivial convergent sequences, every dense subspace of \(X\) is open and every nowhere dense subset of \(X\) is closed and discrete.

**DMS Set Theoretic Topology Seminar**

Feb 07, 2020 02:00 PM

Parker Hall 246

Speaker: **Professor Vladimir Tkachuk**

Title: Any monotonically normal space is discretely generated

Abstract: We will give the proof of the statement in the title and start to construct an example of countable crowded space in which every discrete subset is closed. This example shows, among other things, that not all countable spaces are discretely generated.

**DMS Set Theoretic Topology Seminar**

Jan 31, 2020 02:00 PM

Parker Hall 246

Speaker: **Professor Vladimir Tkachuk**

Title: Some nice classes of discretely generated spaces

Abstract: We will see that it is an immediate consequence of discrete reflexivity of compactness that any compact space is weakly discretely generated. The next step will be to establish that any sequential space is discretely generated as well as any compact space of countable tightness. If time permits, I will also prove that any monotonically normal space is discretely generated.

**DMS Set Theoretic Topology Seminar**

Jan 24, 2020 02:00 PM

Parker Hall 246

Speaker: **Professor Vladimir Tkachuk**

Title: Discrete sets and properties they determine.

Abstract: This talk will be the first one from a series of presentations on topological properties determined by discrete subspaces of a topological space. We will introduce discrete reflexivity together with discrete generability and start proving their basic implications. We will also show how they are placed in the hierarchy of global and convergence properties and indicate some nice discretely reflexive classes of spaces.

**DMS Set Theoretic Topology Seminar**

Dec 04, 2019 02:00 PM

Parker Hall 246

Speaker: **Professor Vladimir Tkachuk**

Title: Point-countable pi-bases in Lindelof Sigma-spaces and some open problems.

Abstract: I will finish the proof that a Lindelof Sigma-space \(X\) has a point-countable pi-base whenever the tightness and the pi-character of \(X\) are countable. After that, if any time is left, I will formulate and discuss some nice published open problems.

**DMS Set Theoretic Topology Seminar**

Nov 20, 2019 02:00 PM

Parker Hall 246

Speaker: **Vladimir Tkachuk**

Title: Any first countable Lindelof Sigma-space has a point-countable pi-base.

Abstract: It is a famous deep result of Shapirovsky that any compact space of countable tightness has a point-countable pi-base. The same result cannot be proved for Lindelof Sigma-spaces because there exist even countable spaces without a countable pi-base. I will show that a Lindelof Sigma-space must have a point-countable pi-base if its tightness and pi-character are countable. This generalizes the above-mentioned theorem of Shapirovsky because any compact space of countable tightness automatically has countable pi-character. It is worth mentioning that this result is not easy to prove even for first countable sigma-compact spaces.

**DMS Set Theoretic Topology Seminar**

Nov 13, 2019 02:00 PM

Parker Hall 246

Speaker:

**Professor Vladimir Tkachuk**

Title: The square of any Lindelof Sigma-space has large discrete subspaces

Abstract: After finishing the proof of Uspensky’s theorem (in which only the cross-star property of compact spaces is missing), I will present a generalization of a Juhasz-Szentmiklossy theorem. Juhasz and Szentmiklossy proved that, for any compact space X, there is a discrete subset D in XxX of cardinality equal to d(X). I will show that the same is true for Lindelof Sigma-spaces.

Last Updated: 09/11/2015