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I have encountered two properties in different areas of math. One is the property (T) of groups and the other is the property (T) of topologies. What is the connection between these two ? Thank you.

Note: A topology on $\tau_Y$ (where $\tau_Y$ is a topology on $Y$) is said to have property (T) if the set $T=\left\{(V,y)\colon y\in V\right\}$ is open in $\tau_Y\times Y.$ The group case is standard. (please see link) A possible relation should come from the Fell topology.

Clarification: A topology on $Y$ is a collection of subsets of $Y$ . A collection of sets may be thought of as a set of sets or, in this case, as a set of elements whose elements are the open sets in $Y$. Therefore, we can define a topology on this set; namely, we can define a topology on the topology of $Y$ . Now $\tau_Y$ is a topological space (as is $Y$) and we can form the product topology on $\tau_Y\times Y$. The topology on $\tau_Y$ is said to have property (T) if $\{(V,y):y∈V\text{ and }V \text{ open in }Y\}$ is an open subset in the product topology on $\tau_Y\times Y$.

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    It is quite unclear what properties you are talking about. If you were more explicit, maybe someone could prove an useful answer.2011-07-08
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    ok I gave the definition.2011-07-08
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    When you say that set is "open on $\tau_Y\times Y$", what do you mean? What is $\tau_Y\times Y$? When you say the "group case is standard", what exactly do you mean? Are you talking about Kazhdan's property (T)?2011-07-08
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    Mariano, open w.rt product topology. Definition uses a new top. on $\tau_Y$ where $tau_Y$ is a top on $Y.$ There is no property (T) for groups other than the Kazhdan's trivial property.2011-07-08
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    What is the topology on $\tau_Y$? Endowed discrete topology on $\tau_Y$?2011-07-08
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    @wxu: You dont need explicitly describe tha topology. It is a collection of subsets of $\tau_Y$ satisfying .... etc. What we know is that it satisfies the above condition.2011-07-08
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    Definition uses *what* new top. on $\tau_Y$? Are you being obscure on purpose? I have never heard about anything by the name of "Kazhdan's trivial property"... Presumably, by asking this question you are interested in that it be understood so as to maximize the chances of it getting an answer that is useful to you. That one has to extract little pieces of incomplete information as to what you actually mean is not exactly a great motivation to give much thought to your question!2011-07-08
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    Can you at least provide a reference where this property (T) of topologies is defined in detail?2011-07-08
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    Mariano: If A topology $\Sigma$ on $\tau$ (which is a set, collection of subsets of $Y$) satisfies the above condition then it is said to have property (T).2011-07-08
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    Ok. I give up. ${}$2011-07-08
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    @Mariano (and others): voting to close as not a real question2011-07-08
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    @Mariano My understanding is: a topology on $Y$ is a collection of subsets of $Y$. A collection of sets may be thought of as a set of sets or, in this case, as a set of elements whose elements are the open sets in $Y$. Therefore, we can define a topology on this set; namely, we can define a topology on the topology of $Y$. Now $\tau_Y$ is a topological space (as is $Y$) and we can form the product topology on $\tau_Y\times Y$. The topology on $\tau_Y$ is said to have property (T) if $\{(V,y):y\in V\text{ and }V\text{ open in }Y\}$ is an open subset in the product topology on $\tau_Y\times Y$.2011-07-08
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    @Ross: I dont understand why it is not a real question ? Question asks the relationships (if there is any) between these two definitions. I will recheck if there is a typo.2011-07-08
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    @categoryboy Could you please confirm that my understanding of the question is correct, i.e., that it is what you have in mind? Could you also elaborate on "the property (T) of groups"?2011-07-08
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    @Amitesh: +1 You are very smart. Thank you.2011-07-08
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    @Ross I think that the question of @categoryboy is valid if he explains "the property (T) of groups" that he has in mind.2011-07-08
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    Agreeing with Amitesh ; there is no explanation of what the property $(T)$ of groups is and it is not obvious in the context.2011-07-08
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    @categoryboy I have edited the question and added my comment as a "clarification"; hopefully the question will then be clearer to others.2011-07-08
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    I just googled it though : the name "property (T)" seems to be a standard. Maybe it should've been mentioned.2011-07-08
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    It is hard in a forum like this to know what to explain and what not. The "property (T)" is unclear to many but may be well known to people in the field. Clearly one should not define $\mathbb{R}$. I would argue it is dense even if not a continuum.2011-07-08
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    The obscurity in the phrasing of this question is staggering. The "group property T" is actually called **Kazhdan's Property T** and is a property not of a group but of a locally compact topological group. The "topological property T" as defined is not a property "of topologies" but of a topology defined on a topology. (Moreover the only motivation given for the question is that they both share the letter T? Wow.) I have a question for the OP: where does this topological property T occur in the literature? For that matter, where do topologies on topologies occur in the literature?2011-07-08
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    @Pete: Motivation is the definition of "Fell topology". Question asks if there is a connection or not. Why does this question caused such a big reaction I don't know. I am not a Shakespeare, people who studied these topic clearly know what I mean. For the references to letter T, goto library.2011-07-08
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    @Ross: My question was important for me but I see that people don't like it. It was doomed to negative reps by a group of people. Therefore, I want to delete this question. Since I couldn't find the delete button, could you please delete or close it. Thank you.2011-07-08
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    @categoyboy: we are trying to understand your question. Please be patient (and, if possible, uninsulting). I'm not attacking you when I asked for references to topological property T: I haven't seen this before, so in order to be of any help I need to read up on it. I looked around a little bit and it seems that there are two apparently different things called the **Fell topology**: one of them is a topology on closed subsets of another topological space (that's pretty close) and another is a topology on the unitary dual of a locally compact group. Is there some relation here, perhaps?2011-07-08
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    @categoryboy There is no need to close/delete this question; in fact, I have voted your question up. I believe it could be an interesting question if we could understand thoroughly what is being asked. I think the lack of understanding of your question says more about the esoteric nature of the relevant concepts than it does about the wording of your question. (I think that your question could have been worded slightly better but that does not mean that it is not a good question.) I disagree with @Ross that this question should be closed and that it is "not a real question".2011-07-08
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    @Amitesh: the question could have been worded *a lot* better, and sharing the letter "T" is not exactly the most convincing of connections. But that there are apparently two things both called "Fell topologies" convinced me too that this *is* a real question. I would now be very interested for someone to explain to me what this property of topologies-on-topologies has to do with representations of locally compact groups...2011-07-08
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    @Pete: Property (T) for groups was introduced by Kazhdan in his thesis in the form "the trivial representation is an isolated point in the Fell topology of the unitary dual (= space of irreducible representations)". It is very important in what is called "rigidity theory" (extending homomorphisms from lattices in a locally compact group to the surrounding group). The Kazhdan-Margulis rigidity theorem asserts roughly that a lattice determines the surrounding Lie group (under higher rank assumptions). However, I've never seen (T) for topological spaces, and a reference would be most welcome.2011-07-08
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    @Theo: right, I am (vaguely) familiar with Kazhdan's property (T) and that instance of the Fell topology.2011-07-08
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    @Pete: I can't say anything intelligent on the relationship of the two incarnations of Fell topologies at the moment except that they were introduced by the same person around the same time for similar reasons and motivations. The Fell topology on the closed sets of a topological space was [introduced by Fell here](http://dx.doi.org/10.1090/S0002-9939-1962-0139135-6) while the Fell topology on the unitary dual appears [here first](http://dx.doi.org/10.4153/CJM-1962-016-6). [This paper](http://dx.doi.org/10.1090/S0002-9947-1960-0146681-0) also seems relevant.2011-07-08
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    @Pete: A quite readable account on property (T) is the [book by Bekka, de la Harpe and Valette](http://books.google.com/books?id=QCftywollBMC), pdf-version [freely available on Bekka's homepage](http://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf).2011-07-08
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    @categoryboy, if it was not obvious, saying "go to a library" is really not an acceptable way to address a request for information. For example, it is clear that *you* could go to a library too: would me telling you that help you at all with your question?2011-07-08
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    categoryboy: I would really appreciate it if you provided at least the definition of the topology on $\tau_Y$ you have in mind. Also, a reference would be nice. Usually the Fell topology is defined on the *closed* sets and I think I see where your question is heading, however I'm not sure. I can safely say that I studied property (T) quite a bit but never stumbled over something like you mention it. Also, I second Mariano's last comment. After all, it's you asking others for help not the other way around.2011-07-08

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